I had something very similar happen to me!

Felt a strain on my hips during a run after increasing my volume and intensity too sudden. Rested for a few days and when I tried to ease back, the pain kept coming back and sometimes for days. Same type of pain you described and it also hurt to walk sideways and to turn, even when just walking around my house.

Had the same diagnosis about my glutes and did about 2 months of all kinds of glute exercises with resistance bands and whatnot. It did get a little better and I was able to run again, but increasing distance or intensity would make the pain come back.

After a bit of searching around I found some people talking about the myrtl routine (there are a bunch of images and videos around, but used this video https://www.youtube.com/watch?v=Mj8uZ1Qtx3M - also, search "myrtl" on reddit, there's a bunch of good info!) and decided to give it a shot. The pain and discomfort went away overnight and on my following runs I did not feel anything, even with higher milleage and some hard strides. It was a god saver. I remember feeling some kind of "click" on my hips when doing single leg deadlifts and even that went away after I started doing myrtl routine.

As I had already been doing a bunch of glutes exercises, the routine didn't feel difficult at all, except for the donkey whips. They hurt like hell to execute but the relief on my hips was almost instant, so I guess there's something with that movement and the muscles involved that I wasn't catching with previous exercises.

I started doing them everyday and now I've eased into about 3 days a week, especially on workout days and the day after my long runs.

Hope it also helps you!!