Do high level sub inputs really sound better?

High level means your amps are putting out the signal for under 20hz....IMHO and IME all amps sound better when not driven to the lowest. I would recommend using low level inputs and dumping the lowest freq. off from your full range amps.
Some subs have a line level XOver...I had very good results with Passive Line Level Xover, say from 50hz up!
You will win a lot of detail and dynamics on the mids and higher freq.
IMHO that is the most important feature of a subwoofer, leaving the lowest Freq. to a speialized sytem (the sub) in order to allow your full range system to breath better on top!

I've heard that using high level inputs bypasses a stage of gain so the sub's amp runs cleaner. This is of course assuming that the amp in the sub is of lesser quality than the main amp, which is often the case. I have a PS Audio GCC amp with remote volume control feeding my subs- I love after all these years of no bass control being able to dial in remotely how much bass I want per song/LP.

I guess it's obvious by now you're not going to get a definitive answer and certainly not a consensus. With my speakers and sub, an HSU, the high level inputs make for a better integration between speakers and sub and that setup is what was recommended by the manufacturer, even though they make provision for using either high or low level inputs.

SImply put, there are advantages/disadvantages to both. Which tradoffs sound better to you?

I have a Velodyne DD15 and it has it's own remote with volume and 5 presets. I used the microphone room tuning and seldom fool with it.