There is nothing to stop one from shooting any of the range sports with any gauge they want. I shoot skeet every week with guys who use .410 or 28 gauge. No one uses 16 gauge because the hulls for reloading are very hard to come by. As to competition limiting ammo availability, I'm sure the manufacturers sell as much if not more field ammo for 12 and 20 gauge. Incidentally, Browning is now making 16 gauge trap ammo and I think Federal recently started doing same.
I think 16 gauge would be an excellent trap gun for ladies. Enough weight to not be whippy like 3" 20 gauge yet with enough punch to reach out there. For same reason I think it would be a great gun for pheasants. Personally, a heavy gun for uplands doesn't bother me that much. Swings better for me. I pack perhaps the heaviest 12 gauge auto ever made (31" Browning A5 Magnum Twelve). It is a chore some days but good exercise which I desperately need. I shoot it very well because it fits very well. I also have Model 12 in sixteen gauge (my very first gun) that I'd love to bring along for pheasants but besides being hard to find ammo, it has fixed full and that won't work on the refuge I sometimes hunt where steel shot is required.