You don't need to abide by a copyright license if you aren't doing anything for which permission of the copyright holder is required in the law -- a copyright license is only needed to do things which would otherwise be prohibitd as within the exclusive purview of the copyright owner.
As the GPL isn't a sale contract that you must agree to as a precondition for receiving a copy of the software, when you receive a copy of GPL-covered software you can do anything you want with it as long as that isn't legally within the exclusive prerogative of the copyright owner (or contrary to the law for some other reason), and the GPL itself is irrelevant. The assertion by the FSF that particular uses of GPL-licensed software are constrained by the license is, therefore, necessarily an assertion that those uses are within the scope of the exclusive rights provided by copyright law.
Ah, that's what I was missing. When did they assert that the use by itself (with no redistribution) is constrained by the GPL? That seems to go directly against their FAQ:
If I only make copies of a GPL-covered program and run them, without distributing or
conveying them to others, what does the license require of me?
Nothing. The GPL does not place any conditions on this activity.
The question at hand is whether you can impose these restrictions in the viral fashion of the GPL, e.g., on code I write that merely makes use of your APIs.
The way that the GPL attempts to enforce this is with copyright law, by not granting you the right to redistribute GPLed software unless you comply with it. By design, it doesn't restrict anything else about how you use the software.
I mean, could you have a clause in a license that said "to distribute this work, you need to wear purple shoes on Mondays"? If so, why wouldn't you just be able to say "to distribute this work, you need to distribute works that link to it under the same license", as as arbitrary condition?