Doesn't Bret Contreras consider the hip thrust and the kb swing to both be horizontal hip extension exercises? That would make the bridge just a primer for the main movement, the swing, and would make weighted bridges unnecessary.
Not sure I'd feel comfortable doing these weighted, but then I've never tried. They are a solid means of isolating the glute, and working it to full extension, so doing them with a sandbag etc across the hips might be a good move.
For me they were a finisher at the end of a circuit to isolate the glute and promote some hip mobility. I have these in the same role on a circuit I wrote up for the wife.
The Glutes are the biggest and strongest muscle in the body. At least they should be, most peoples butts are dysfunctional and underdeveloped. Especially men.
When I trained Weightlifting we would use Explosive Barbell Hip Thrusts to get the hips bumping. I used bodyweight for explosive sets of 3-5. If used at the end of training for strength we would train Barbell Hip Thrusts with heavy weight, 2-3x BW, at strength or hypertrophy protocols. A few Heavy Weights and Super Heavy weights I trained with did Hip Thrusts with 700-800lbs, and had over 300kg totals.
The Glutes can handle a lot of work.
Do Barbell Hip Thrusts before Bench Press and see big gains in Leg Drive and strength. Do them after Deadlifts and Squats to thoroughly train the Glutes.
As you can probably tell I like Barbell Hip Thrusts.