I know that NEMA 14-60 plugs and receptacles exist, but they seem really rare. I've come across a few random online forums where people say that this is because they're not allowed by code, since any load of more than 50 amps must be hardwired in residential settings. Nobody has linked to a source for this claim, though. Is this the case under any version(s) of the NEC that are in use in the United States today? If so, where exactly in it is the prohibition?
You can have a 14-60R in your house, but it must be on a dedicated branch circuit
There is nothing in the NEC that prohibits you from having a 60A (or larger!) receptacle in your house. (While NEMA-style receptacles and plugs only go up to 60A, it is possible to get higher amp ratings in the form of industrial-style pin-and-sleeve connectors.) However, NEC 210.18 requires that receptacle to be on a dedicated (individual) branch circuit (the exception's only useful in factories, so it's not mentioned here):
210.18 Rating. Branch circuits recognized by this article shall be rated in accordance with the maximum permitted ampere rating or setting of the overcurrent device. The rating for other than individual branch circuits shall be 15, 20, 30, 40, and 50 amperes. Where conductors of higher ampacity are used for any reason, the ampere rating or setting of the specified overcurrent device shall determine the circuit rating.
On top of the very good answer from @ThreePhaseEel, I'd add:
You should probably reach out ahead of time to your local inspection agency, to see what stipulations they'll put on it. They may look into whether the receptacle's safety approvals are "for domestic use" as opposed to a commercial/industrial category. And they may become very concerned that you might be planning to backfeed a generator via a double-male cord, which a disappointing number of homeowners actually try to do. Even if you can show a legitimate use, they may fret that a future owner (or future you) will try it at a moment of crisis. Basically a very-high-amp plug connection into a home's circuits is a red flag for them.
And of course your insurer will want to know about it before you file a claim. They may have concerns not just about the installation of the circuit but whatever it is you're doing with it. If you're welding or running a kiln, you're generating a lot of heat, you may be doing it as a commercial activity, etc. You want all this documented on your policies ahead of time.
If it's for something like a vehicle charger that you're retrofitting with a plug, the insurers and the inspectors may (should) flag that it's not the manufacturer's configuration, which can break your insurance and your approval on top of your warranty. (And if the charger has backfeed capabilities, then please put the receptacle down and head directly to the transfer switch aisle.)