I was running iPadOS 16.0 beta 3 with developer mode enabled and all was good. For reasons that are not important to this post, I had to do an "Erase All Contents and Settings". After the OS reinstall, I signed-in with the same Apple Id as before, I re-downloaded the beta profile from Apple, and went to enable "developer mode" (which requires a reboot). However, upon restart, developer mode remains disabled. I tried several times, but developer mode just won't turn on.
Now I'm left with no device to test iOS16 apps :-(
I solved the problem. I erased the iPad again. When I went to installed the beta profile, I realized that before I may have installed the wrong profile (iOS instead of iPadOS).
In both cases, the device was already in iPadOS beta 3. However, the second time, after I installed the right profile I was offered to download an update. I accepted and after that I was able to enable developer mode.