I double batt quite often but I quilt on a longarm. I think it would present a lot of challenges to quilt a double batted quilt on any stationary machine (including extended throats) but even more so on a domestic. You are already dealing with a lot of bulk with single batting. I imagine it would present challenges in making the sandwich as well. Maybe someone who quilts on a sitdown machine can chime in with their views.
On a stand up longarm I have never had any issues. Combinations I have used are wool over 80/20 or W&N, Hobbs Polydown over 80/20 and this quilt was double batted poly.
Orange and blue Kaleidoscope circle quilt "Tropical Parrotise". And this quilt had trapunto treatment and double wool so under the applique of the goddess is 3 layers of wool batting.
Quan Yin - A show quilt. And this quilt I did a layer of Hobbs Polydown over an old wool army blanket.
Batik block swap quilt
Double batting does effect the quilt's drape. It also makes for a very warm quilt and depending on what combination of battings you use can also be very heavy. The batik one with poly over the wool army blanket is super heavy and I can only use it as a bed quilt in the very dead of winter, like those -30 nights. For use on a bed, I feel the lack of "scrunchability" (lack of drape) does not have any impact. The quilt still hangs nicely over the side of the bed but if you want a quilt to cocoon into and scrunch up all around you, I would not double batt.
Double batting really emphasizes quilting motifs, provided you quilt closely enough in the background. It often gives a trapunto look and that is why many show quilters do it, myself included.