I did indeed run a BBS on OS/2 and played X-Wing vs. Tie-Fighter while people were dialing in without a hitch.
At least that's how I want to remember it. Surprisingly I never tried DesqView but it looks great!
I ended up running OS/2 Warp when it came time to run my own BBS. :)
OS/2 had great multitasking, but I remember its driver support being limited for multiport serial boards. I had an Arnet Smartport 8 and couldn't move from DesqView to OS/2 because there was no support for it.
I think they leveraged this to do stuff like update the appearance of buttons on click in the higher priority event process (and then send an event), to give a more responsive feel to the UI.
I also remember Mosaic once had a use after free bug on graphics contexts and would occasionally render a web page in another window.