So I have a world where end gateways don't spawn after you kill the dragon. I think I created the world in 1.16. When I killed the dragon for the first time no end gateways appeared. I beat it the second time and again no gateway, so I had to bridge a thousand blocks to get to the outer end islands. Recently I opened the world in 1.17.1 and defeated the Ender dragon the third time, but still no gateways. In other worlds they do generate, so I'm a bit confused why they don't spawn in this world.
You mention you created the world in 1.16. Is it possible you created it in one of the snapshots for 1.16? If so then the cause of your problem is possibly a result of a bugged 1.16 snapshot that broke end gateway generation for any world opened in it; see MC-184482 for a way to fix.