I definitely prefer poly threads over cotton threads. But I quilt on a LA so my reasons are probably different than many others. My LA speed is much faster than a DSM so a cotton thread will produce much more lint than in a domestic.
There are many more choices with Poly threads from 100 wt which is as fine as silk all the way up to 12 wt (which is the heaviest I can run through my top side. I can run 8 wt if I load it in the bobbin).
Like Prism, I find a poly thread allows more drape with dense quilting. I love the shine of a poly along with the intense colors.
There are many quilt purists out there who don't want any synthetics in their quilts at all. (although it is pretty hard to find cotton batting with no scrim and even if you do, the recommend minimum quilting distance is 2"). So many shun poly threads for that reason.
There is also the old wives tale about poly thread "cutting" through the cotton fibers of the quilt over time. Which is not true at all with today's wonderful polyester threads. But that doesn't mean you can ignore weight, ply, etc. You may need to experiment with some different polyester threads.
I have used Isacord, Glide and Invisifil all with great success. I have quilted very densely and still had great drape but didn't need it as the items were wall hangings.
I think batting will impact the drape more than the thread used for quilting will, unless you have laid down so much thread it was more akin to thread painting than quilting.