"which university do you want to attend" was rejected but I think it's ok
How about "go to" as a synonym for "attend"?
that would work
"Which university would you like to go?" was rejected but I think it is acceptable
We can't say 'go' without 'to' in this context.
It's interesting that Duolingo uses ambiguous, region specific terms like college. Why not university? Afaik that means the same thing all over the Anglophone world.