Short answer: 1. Yes. 2. The meaning comes from its words.
To get (a)round to is a phrasal verb: The origin of the meaning is simply found in its words.
The Oxford English Dictionary defines both to get around to and to get round to as phrasal verbs, having the same meaning:
With to: to succeed in finding the time, energy, or inclination for (doing something); to come to the point of dealing with.
Without the to, the verbs diverge in meaning, as you can see at the end of this post.
to get around to
The earliest recorded usage in the OED is from
1887 M. E. Wilkins Humble Romance 35 There has been a good many things I haven't got around to.
followed by P.G. Wodehouse in Laughing Gas:
1936 P. G. Wodehouse Laughing Gas xxvi. 269 ‘I want to know why you haven't tied him up.’.. ‘We was aiming to get around to it later.’
which is an interesting quote also because of was aiming to, which means
To have (something) as an object, intention, or desired outcome; to be determined upon; to seek to achieve or obtain.
(a) With infinitive as object. Also simply: to intend, to mean (formerly chiefly England regional and U.S., now colloquial).
The first usage is from 1400, so this goes way back. But it used to be a regional thing; it is now considered colloquial.
Before WWII Wodehouse lived in both the US and England; he often represented "the English" to American readers and vice versa, publishing books in both countries. Wodehouse had a 75-year career as an author. Astonshing. Even in some of his later works he uses expressions found in his earlier works 70 years later, when the world had been a different place (before both world wars).
The first genuine usage for get round to is 1946.
1946 K. Tennant Lost Haven (1947) xiv. 221 Everything in Lost Haven was put off until someone should have enough time to ‘get round to it’.
Notice the quotation marks around the phrase; this indicates something about the attitude of the phrase by the narrator of the story. Perhap he means the phrase is new-ish or that its usage is ironic. (By the way, Wodehouse has a lot of uses of words with the ending -ish, showing that this suffix is not anything new, despite its recent resurgence in use.)
As Jasper has shown, we can use can before get around to, just like we can many verbs, with the sense of being less direct or more polite. To see this meaning of can, try it as an imperative (an order):
You can sit over there. You can sit here.
These are actually orders, but the can significantly softens them.
So, you know
Why don't you upvote this answer and Jasper's, whenever you can get around to it? :)
Without the to, the two phrasal verbs have other meanings:
to get around can mean
To go round; to circulate; specifically to go out a great deal; to visit many places. Chiefly U.S.
1928 Amer. Speech 3 219 Get around, to..have many desirable dates. ‘Mary Jane sure did get around last semester.’
Note that today if we said "Mary Jane sure gets around," it would often (always?) implies that she sleeps around (has sex with multiple partners, or is at least alleged to). So you can see how the meaning has changed from 1928 to today.
1951 M. McLuhan Mech. Bride 60/2 The news got around fast.
1959 Times Lit. Suppl. 20 Mar. 159/3 Still, Mr. Donnelly has got around... He makes his way to places like Tashkent, Samarkand and Alma Ata.
to get round can mean:
To recover from illness, get well.
The idea, I imagine, is to get round a disease and make it back to health. But I can't recall this phrase actually used this way. Nowadays, we say come (a)round (apparently only an AmE usage) or, using a different verb pull through.
1857 T. Hughes Tom Brown's School Days ii. vi. 349 Did they tell you..that poor Thompson died last week? The other three boys are getting quite round, like you.
I wouldn't use this phrase to have this meaning, as it seems outdated, although the OED does not indicate it as such. But come around is used today, as is pull through.
Best Answer
In this case, "sticking ~ on" doesn't mean turning on in the electrical power sense, but placing on top of: to operate a kettle (a vessel for boiling water for beverages) one puts it on a stove and then turns the stove on.
The author is creatively alluding to making oneself a cup of tea as a break ("and spending a moment or two"), and proposing now is a good time to do it -- a way of indicating coming to a conclusion of a conceptual chunk of the text, like a chapter, and suggesting taking that break as a kind of reward ("reflecting on how far you’ve come. You’re halfway to being a professional!") for all one's hard work so far.
The technical term for this particular sort of poetic turn of phrase is metonymy -- he's using "kettle" to refer to making oneself a cup of tea. Or even we could say it's a metonym twice removed: the hypothetical tea alluded to could, itself, be described as a metonym for taking a celebratory respite in one's labors.