Results from a Google Books search strongly suggest that the expression originated (in print) in religious publications. At least in my search results, publications of the missionary wing of the Methodist Episcopal Church account for the first five matches, spread across a period of seven years. The earliest of these sources (from January 1910) credits the saying to a Commissioner McFarland, who may be the author of the third article cited below.
From "Never Paralleled in New York," in The [New York] Christian Advocate (January 20, 1910):
Many ministers have heartily indorsed the [Laymen's Missionary] Movement and will support it with their indispensable influence. With their cooperation failure is impossible, and it is the universal post-convention report from the other cities that all local causes have been helped on by the new impulse of the Movement whose watchword is "For the Other Man." As Commissioner McFarland said at the Astor dinner, The rising tide lifts every boat!
From "Japanese Church in Wonsan, Korea," in The [Nashville, Tennessee] Missionary Voice (May 1911):
Now the building of the church in Wonsan [Korea], of course, was not entirely, or maybe even mainly, responsible for the building of the great church at home. But at least it did not hinder, and who doubts that the enlargement of heart that came to them through that unselfish thing, the spiritual swing developed in that enterprise abroad, hastened and helped on the larger thing they would do at home? "The rising tide lifts all boats," and "the light that shines farthest shines brightest at home."
From Henry B. F. McFarland, "The Man by Man Rise of a Race of Men," in Association Men (January 1915):
The rising tide lifts all the boats upon it. All parts of the colored [Young Men's Christian] Association movement have shared in the new progress. Great corporations employing negro workmen—like the Newport News Ship Building Company, which has already started work, with its four thousand negro employees—see the advantage of such Associations.
From "Woman's Home Missionary Society," in Minutes of the One Hundred and Seventeenth Session of the New York Conference of the Methodist Episcopal Church held in First Church, Peekskill, N.Y. March 22, 1916 (1916):
Wherever there is an active, interested, wide-awake Woman's Missionary Organization there will be found a broadened vision, an increasing zeal along all lines, and a permeating spirit of the Christ-love that is most helpful. "The rising tide lifts all boats."
From Hugh Burleson, "The Progress of the Kingdom," in The Spirit of the Missions (May 1916):
The editor of THE SPIRIT OF THE MISSIONS is certain that he speaks not only for himself but for every official connected with any missionary enterprise when he declares that he would not, if he could, divert a single dollar from the relief of Europe. On the contrary, he rejoices when gifts for the war sufferers increase, not only because of his passionate desire that this awful suffering may be relieved, but also because he knows that "a rising tide lifts all the boats," and that every good cause should profit by the sympathy which this crying need awakens in the hearts of men and women who too long have been concerned chiefly about their own comfort and gratification.
From "A Hint from Harvard," in Harvard Alumni Bulletin (December 11, 1919):
There must be hundreds of Harvard men who on reading that [story] will directly double their subscriptions; the tale is so darned humane! I am not a Harvard man as you know, but the Top Sergeant and his wife make me double my paltry ten dollars. The psychology of the outsiders with colleges of their own to serve is quite correct. A rising tide lifts all the boats!
And finally, the cover of American Gas Monthly (March 1920) consists of the following quotation and subtitle:
"The Rising Tide Lifts All the Boats."
When the tide of public opinion swells through recognition of service well performed, all our boats will be lifted.
Basically what we have here is a slogan so successful that in a single decade it went from being a watchword—indeed almost a catchphrase—of a vigorous Christian missionary movement to becoming an inspirational proverb for members of the American Gas Association to contemplate.
A Google Books search for "professional bias" for the period 1700–1800 yields four legitimate matches—all of them connected to religion. From Richard Watson, An Apology for Christianity in a series of Letters, Addressed to Edward Gibbon, Esq (1777):
I beg pardon for styling their [the Deists'] reasoning, prejudice ; I have no design to give offence by that word ; they may, with equal right, throw the same imputation upon mine ; and I think it just as illiberal in Divines, to attribute the scepticism of every Deist to wilful infidelity; as it is in the Deist, to refer the faith of every Divine to professional bias.
Bishop Watson's response to Gibbon is cited in Letter 44 of Richard Sullivan, A View of Nature, in Letters to a Traveller Among the Alps (1794):
A latent, and even involuntary scepticism, certainly adheres to some characters. And therefore, it is illiberal in the advocates of religion, to attribute the scepticism of every Deist to perverse infidelity ; as it is in the Deists, to refer the faith of every Christian to professional bias.* This particular bent we can neither comprehend, nor estimate.
*Bishop Watson.
As Hot Licks notes in a comment below, this is by no means an independent occurrence of "professional bias," but rather a restatement of the previous instance, with Christian substituted for Divine.
From a letter to the Philological Society of London by N. N. on March 7, 1787, in The European Magazine (March 1787):
We have a hint also of "the number and ability of unbelievers." I will not class the Reviewer with those Free-thinkers, as they call themselves, who are mere slaves to the opinion of others ; though I suspect him to have very little knowledge of the facts or answers in defence of Christianity. With those, however, who disbelieve, not from any reason they themselves can give, but because some acquaintance of theirs, of whom they have a good opinion, or some celebrated writer, as Voltaire, Hume, disbelieved, we may argue in their own way, and confront them with names and authority, I trust, superior to any they can produce. ... To say nothing therefore of the bulk of the community, high and low, rich and poor, learned and illiterate, which or so many ages have believed in the Gospel, let us only urge the names of Mede, Cudworth, Barrow, Clarke, Jortin, ; of Leland, Taylor, Lardner ; of Le Clerc, Limborch, Mosheim ; men who spent whole lives in the study of Christianity, and manifested as much freedom and acuteness in their researches, as are to be found in any science whatever. Let us add the authority of Bacon, Grotius, Locke, Newton, Hartley, men who were under no professional bias, and did not take their religion upon trust, but each of them spent many years in inquiries into it, and rose up from the inquiry fully and firmly persuaded of its truth.
And from The Parliamentary Register (April 28, 1795):
The Bishop of ROCHESTER disclaimed having any professional bias : he said, in all great bodies of men there were some undeserving objects, but it would be unjust to punish the worthy on their account.
As two of the four eighteenth-century instances of "professional bias" in the Google Books search results are from bishops, a third quotes one of the first two, and the fourth distinguishes between "men who spent whole lives in the study of Christianity" and men who merely "spent many years in inquiries into it" (arguing that the latter did not have a professional bias), it seems clear that prior to the 1800s the term was understood to refer to religious profession.
The earliest nonreligious (or religion-neutral) instance of "professional bias" in a Google Books search appears in a review of John Fuller, M.D., The History of Berwick upon Tweed, in The Monthly Review (October 1800):
This design [to improve "the present state of agriculture and commerce of his native town" and to propose "the real happiness of the inhabitants"] is no doubt truly benevolent and patriotic : but surely it was not necessary, in order to impress on the reader's mind the importance and utility of agriculture, to give an account of man in a savage state ; nor to present us with various other observations which here occur, and which seem to originate in the professional bias of the author's ideas.
Since the author is a medical doctor and not a divine, I assume that his professional bias is in the direction of modern medicine. The review doesn't mention religion at all.
This rather limited record supports the idea that "professional bias" began as a term connected to the profession (that is to say, the professing) of Christianity and used by various religious and nonreligious writers, and that from there it expanded to include professional occupations or livelihoods within which a particular viewpoint or presumption or interpretive inclination predominates.
Best Answer
The original idiomatic use of 'stay the course'
As phoog mentions in a comment above, the earliest instances of "stay the course" meant "to halt or temporarily impede a process." Thus, Ben Jonson, Volpone, or The Fox (1605) has this line:
From "Certain Observations touching the two great Offices of the Seneschalsey or High-Stewardship, and High Constableship of England" (1641), in A Collection of Scarce and Valuable Tracts on the Most Entertaining Subjects, volume 5 (1811):
Two examples from the early nineteenth century involve ships and their progress through the sea. From Thomas Maurice, Indian Antiquities (1812):
And from Richard Pering, A Treatise on the Anchor ... and a Schedule of Proportionate Weights of Anchors (1819):
All the pretty horses
The use of "stay the course" in the very different sense of a horse completing a race appears for the first time in Google Books search results going back to 1885, in a citation in Farmer & Henley Slang and Its Analogues, volume 6 (1903):
By 1902, the phrase "stay the course" seems to have become thoroughly established in horse-racing lingo, but to have broadened to mean something like "compete successfully and finish well. From "The World of Sport," in The Sketch: A Journal of Art and Actuality (August 27, 1902):
'Don't halt the course; hold to it!'
This was followed surprisingly quickly by yet another sense of "stay the course"—one very nearly diametrically opposed to the sense that prevailed between 1600 and the late 1800s: to continue on a particular course of action regardless of unexpected setbacks encountered along the way. The first instance appears to be in "Military Service Bill, Second Reading" (4 May 1916), in The Parliamentary Debates (official Report): House of Commons, third volume of Session 1916 (1916), which rather oddly shifts into that sense of "stay the course" from the racing sense:
And in Minutes Taken Before the Royal Commission on the Civil Service (1929–31) (31 January 1930) [combined snippets]:
Here the notion of "stay the course" means (as it seems to have done in the speech about financing World War I) "sticking it out" or "seeing it through"—the modern meaning of the phrase.
But instances of the old usage continue to occur at least as late as 1980. For example, from Maurice Collis, Foreign Mud: Being an Account of the Opium Imbroglio at Canton in the 1830's (1946):
And from Henry Dodwell, The Founder of Modern Egypt: A Study of Muhammad Ali (1980):
Interestingly, China Notes, volumes 14–23, pages 55 & 89 (1979[?]) [combined snippets] uses the phrase twice, in opposing senses, within 35 pages of one another:
The triumph of holding steady
In the United States, the decisive moment when the public embraced the (relatively) new meaning of "stay the course" and abandoned the old one seems to have occurred in 1982. In the off-year Congressional election (a federal election during a year when the Presidency was not being contested) in 1982, President Reagan, two years into his first term in office, made "Stay the Course" his rallying cry for electing Republicans to Congress to support his economic policies, which had run into some turbulence. The last gasp of the old understanding of "stay the course" came in this rather airily delivered note in Verbatim, volume 9 (1982) [combined snippets]:
But that's the funny thing about idioms, isn't it? In the event, the Verbatim writer was powerless to stay the course of linguistic change, as English speakers chose not to stay the course with the original meaning of "stay the course."