Speaking in very broad, generalized, terms, I tend to find pushpull amps to have a "tighter" more punchy sound than SET amps. But, too much of that quality and the sound seems a bit artificial or "mechanical" and less natural than the more relaxed sound of good SET amps. For me, for example, KT88 pushpull amps tend to be particularly likely to sound too punchy and mechanical for my taste, but, as with everything in audio, there are always exceptions.
I agree- I tend to see punchy bass as a coloration, since you never seem to run into it in live events. I've played in orchestras, jazz and folk ensembles, rock bands and just not encountered it. I think its the product of an overdamped speaker to a certain degree but I don't think that's all of it.
The distortion signature is what is at play here IMO. Most pentode-based amps employ a bit of feedback, but since they really don't have enough, they have an obvious distortion signature (the amplifier's 'sonic signature'). Some of that includes harmonics of bass fundamental tones, generated by the feedback itself. The ear is interpreting them as 'punchy' (and yes, this happens with many solid state amps too). IMO/IME the way around this is to make sure the amp is as linear as possible, so much so that it will sound reasonably decent with no feedback at all. Only then do you apply the feedback- and then it should be used to control gain more than anything else. With a pentode-based amp I suspect this would mean that the output section is wired Ultra-Linear.
But many pentode output stages really aren't that linear and need the feedback to linearize them. That is a bad move IMO. You can't get rid of all the distortion and if you start with a high distortion amp, the feedback will cause it to have a less musical distortion signature overall when you're done. Amps that have zero feedback have a different distortion signature that tends to favor the lower ordered harmonics which will mask the presence of the higher orders. But if you have no feedback you also have a much more limited range of speakers that are practical with the amp. For more on this see
The Power Paradigm Vs the Voltage Paradigm