Bad connections are equally evil for any sort of wiring. Yes, old buildings..... I've lived in them too. I grew up in a house built in 1914, with knob and tube 14 ga wiring, and live in a house built in 1933. No knob and tube, but plenty of 14 ga, in that delightful old cloth covered hardened rubber insulation.
Scariest thing in the 1933 house was an added circuit, wired with 14 ga aluminum wire, tied into a 40A stove circuit. I disconnected that virtually instantly, and did not pull a permit to do it, either. (there was a double 120V outlet wired in by the electrical panel, in addition to the one in the dining room that it served)
As for "throttling current", well, sort of. If you draw current within the limits of the wire gauge, it's fine. The issue is a voltage drop. At 20A, the drop is not that much added, 133% of normal max drop. And a bit less than double the usual heating.
If a 3% drop is taken as maximum, and you were at that already with 15A, then you would have a 4% drop at 20A. A drop of 4.8V instead of 3.6A.
It would certainly put a limit on fault current, if a dead short existed at the end of the wire. The max current with 14 ga would be less than with 12ga.
Might it, for instance, have some effect on charging of the bus capacitors in a VFD? "Some", yes.
Would that be significant vs 12 ga? That depends on the effect of 1/3 more resistance.
However, the most worrying issue with trying to run a larger motor in an old apartment, is what you and several of us already mentioned....... Is the wiring done correctly, does the breaker and/or outlet correctly represent the circuit capacity?
The outlet is no guide at all, 20A outlets are installed by landlords willy-nilly. The breaker SHOULD be a guide, but that depends if the box and breaker were installed by an electrician, or by the brother-in-law.