Could you send over your recent server log as any permission issues will be logged within there?
Also, it might be worth opening the dev tools on your browser and checking to see what call is hanging up. That will let us know what might be happening.
One more quick thought while you are testing, you may try creating a test user against the built-in Runbook Consumer role to see if the page will populate as there are a few differences in that role versus the screenshot for the custom role. With the prompted variable, my best guess is that you may need the LibraryVariableSetView. However, as Sean mentioned previously, it should generate a permissions error that shows up in the Octopus Server logs.
I just wanted to jump in quickly as well as I have set up a user locally with only those specific permissions granted to it, and the user was able to run a runbook after setting a value to a prompted variable in it. An additional thing worth checking I think would be the scoping of the user role in the team, or if this user is part of any other teams that might be restricting permissions unexpectedly. If you go in your web portal under Configuration > Test Permissions and select this user, you can export this to .csv, which we’d be happy to have a look into.
Another potentially important piece of information - could you let us know the Octopus server version you’re running?