Here are the steps that solved it:

(1) Voodoo band on quads and hammies, all the way up onto the hip.  I also voodoo band my foot and calf, but I don’t think that had a material impact on this problem (but I could be wrong).

(2) Using a lacrosse ball, golf ball, and the pointy end of the Gemini, to dig into the tender spots in my hips and glutes.

(3) Wrapping a band around my thighs, just above the knees, and doing side steps

Note that (1) and (2) had to be repeated many times until things really loosened up.  I now do them both on a regular basis as a kind of general maintenance.

I specifically felt something pop in my left hip after doing (2) and (3) above last week.  Something felt right about it, so I tested my deadlift, and sure enough I was lifting 60-70 lbs more, and all the pain was gone.  Nothing was gradual about this — for years it was 345 feeling like a 1-rep max and just feeling wrong, and then suddenly 405 went up relatively easy, with no pain, with no weight belt.

I tested the squat a few days later.  I’ve never felt so strong squatting deep.  My back stays engaged very low in the squat, which I was never able to do before.  It had nothing to do with just focusing on engaging the muscles, or improving my form (I mention that since a number of coaches thought the problem was along those lines).  Once the quads, hammies, glutes, and hips were loosened up, it was very easy to stay tight in the bottom of the squat, and once again, lift heavy with no belt and no pain.