The kitchen sink faucet came from the Home Depot Glacier line:

A “FastMount” push-and-turn nut secures the central pillar to the deck, although it had worked loose since it was installed some years ago. After retightening the nut, however, the faucet spout and handle remained loose, which I eventually figured out was due to the central pillar having worked loose from the plastic body inside the spout.
The solution involved releasing the FastMount nut, pulling the whole affair out of the deck, and tightening the threaded pillar into the body. After a few false starts, I applied a pair of grippy leather gloves and a firm grasp to twist the pillar another quarter turn into the body, after which it installed properly:

The mark on the far left shows how much I twisted the pillar:

If I don’t tell anybody, they’ll never notice.
I fear this will not be permanent, in which case I must disconnect all the plumbing, take the faucet into the Basement Shop, and have my way with it. Most likely this will involve thread locking compound applied to parts that aren’t visible without a complete disassembly.
For the record, the setscrew securing the faucet handle to the valve fits a 2.5 mm hex wrench, aligned just about parallel to the handle rather than perpendicular to the rear surface:

In comparison to the never-sufficiently-to-be-damned American Standard faucet in our previous kitchen, the spout does not depend on rotating O-ring seals, because the valve sends water to the integral sprayer through a flexible hose. Although the spout does have an O-ring at the bottom, it serves to keep casual splashes out, rather than pressurized water in.
With a bit of luck, this will work forevermore.
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