Have you ever seen a new-fangled bass that's square in cross-section? They seem to play just as well and, I suppose they would be easier to fabricate.

They're not all that new. The first one was made by Alec Loretto in the 70s, written up in a recorder magazine at the time as a DIY project readers could try. Paetzold put them into production a bit later, probably independently.

I've heard one of the Paetzold ones. It was terrible. Turned the owner had assembled it wrong in some way, but i haven't heard her playing it since she figured that out.

For playing position - whatever works; there are pictures from the 18th century of players holding the bass recorder as you describe. I prefer vertical, since my Zen-On is so light I can take all the weight on my right thumb and not use the neckstrap, if it's for a short piece. For my greatbass, I usually play sitting down, balancing the bell between my feet.