Kate Gregory has outlined the answer, but perhaps more detail is necessary for non-native speakers to understand the underlying assumptions.
First, in the Time article the writer is using an extended metaphor sometimes referred to as a "literary conceit":
In literature, a conceit is an extended metaphor with a complex logic that governs a poetic passage or entire poem. By juxtaposing, usurping and manipulating images and ideas in surprising ways, a conceit invites the reader into a more sophisticated understanding of an object of comparison.
The "more sophisticated understanding" we are being invited into is one involving standard, commonplace tropes from romantic breakups.
First, the title refers to the standard plain-truth version for ending a relationship that is usually given by a third-party to the person being jettisoned:
"He [or she] is just not that into you."
In other words, it's over. Get past your issues with the breakup and move on with the rest of your life. In the context of banking, banks are being told by the third party (Time) that consumers really don't care about them anymore.
Here's where Kate's answer enters the picture. The standard trope "It's not you, it's me" — used to absolve the other party in the relationship of blame as a way of pre-empting any desperate attempts to salvage it: "But I can change!", "We can work it out!", and so on. By taking responsibility for the breakup there is no possible comeback. The person on the receiving end of the breakup is left without recourse.
Now, we have to address the fact that this trope is inverted: "It's not me, it's you." That plays the trope for laughs, since instead of absolving the jettisoned party of responsibility, the person broken up with is saddled with all of it. In this case, the banks are being seen by consumers as bearing all the guilt for the relationship falling apart, and are (by refusing to use banking services) being told so by them.
By using such devices, the writer presents the information in a light, clever style that is not strictly suited to the bald representation of boring facts about banking, but which is no doubt much more readable for the same reasons.
Best Answer
The term "cut checks" goes back more than 150 years, but the usage is complicated by the fact that in some instances "cut checks" signified "canceled checks" while in other it signified "created checks." The allied term "cutting checks" goes back at least 147 years.
'Cut checks' as canceled checks
The earliest instance of "cut checks" (in the sense of physically canceled checks) in Google Books search results appears in Fraudulent Canal Scrip: Report of Evidence before the [Illinois State] Senate Committee on Finance (February 4–15, 1859):
From People of the State of New York v. Morris Speigel (December 12, 1892), reprinted in New York Court of Appeals. Records and Briefs (1894):
From Montague Glass, "The Center of Population," in Munsey's Magazine (August 1908):
And from Alden Moore, It's Your Business: Your Use of Your Money to Buy Better (1949) [combined snippets]:
Further complicating this sense of "cut checks" is the fact that in a patent application for "Ticket-Boxes" filed on June 11, 1877, the inventor refer to the device as containing "a machine for cutting pieces or checks from tickets," suggesting that "checks" in that instance referred to something similar to "chads"—and not to "bank checks" at all.
'Cut checks' as created checks
The earliest instance of "cut checks" in the sense of "create or make out checks" that a Google Books search finds is from George Jackson, A New Check Journal, upon the Principle of Double Entry, sixth edition (1841):
From United Sates Patent Office, Annual Report of the Commissioner of Patents for the Year 1870, volume 2 (1872):
From The Inland Merchant, volume 2 (1910) [combined snippets]:
From Schnipper v. Township of North Bergen (New Jersey Superior Court, decided April 11, 1951), reprinted in Reports of Cases Argued and Determined in the Superior Court, Appellate Division, Chancery Division, Law Division, and in the County Courts of the State of New Jersey (1951):
From what appears to be a caption in The Punched Card Machine Accounting and Data Processing Semi-annual (1952) [snippet view]:
Also relevant is an ad for a hydraulic paper cutter designed to meet banking needs. The ad ran in both Burroughs Clearing House, volume 44 (1959), and Banking: Journal of the American Bankers Association, volume 52 (1960) [combined snippets]:
Conclusions
Examples from Google Books indicate that "cut checks" goes back to the nineteenth century, both in the sense of "create or make out a check" and in the sense of "cancel or spindle a check." From the discussion in George Jackson's 1841 book on accounting, it appears that in that era checks were indeed cut from a master checkbook at need. Advertisements for automated check-cutting machinery have appeared as recently as 1960, suggesting that physical cutting of checks continued to occur (in some settings) more than a century after Jackson addressed the topic.