So I know nothing about this subject and am interesting in what scenarios the amplifier current capacity matters, and when or if the issues that it solves are ever audible?
I know just enough to have bought an amp where I will never have to think about this again. Purifi ET400A based.
As a practical matter, you can just look at your speakers (8 or 6 or 4 ohm) and make sure the amp is rated for that resistance. The typical amp ratings are for 8ohm, and if they don't list 4 ohm watts, I would avoid that amp for a 4 ohm speaker. Amp specs are usually given for 8 ohm loads up front, but looking for what they do for 4ohm might take more looking. As a rule of thumb, using the marketing speak, I would say that a 100w/8ohm amp that goes to 200w/4ohm is "high current". If it only goes up to 150w at 4ohms, I would not call that high current. Definitely not as high, that's for sure.
The lower the resistance, the more important current becomes.
What it sounds like is distortion, particularly in the bass. Things might sound flat, dynamic passages might not be dynamic enough. So those are the main issues more current can solve, if that's the cause and not something else.
If I wanted to demonstrate how this sounds to someone in the room, I would use my headphones, which are Hifiman XS (one of the best deals in audio I would say). Not terribly hard to drive for planar magnetic drivers, but they do benefit from some oomph.
Play on my phone with a $20 dongle, flat, play with my $300 AVR headphone jack, flat. Dull, sounded no better than my $20 IEMs either way. And I never listen to headphones loud, but I was still cranking the volume pretty high for those two amps.
With a Topping Dx3 pro? WAY better! More current, more oomph, much clearer sound. At the same volume (close to it, by ear not by dB).
That's my practical, not very technical take on the issue, fwiw.