Regardless of what happens with other "What" questions or "that" clauses, there is simply no way to ask your original question without "of". Even though we can use "We were afraid that ..." in the answer, that can't be converted into "What were you afraid that?" or your shorter version. There is only one grammatical question there:
What were you afraid of? (You might hear people of a much older generation use "Of what were you afraid?")
In the response, what follows "afraid" can differ:
We were afraid of the dark.
We were afraid [that] he would find us.
We were afraid of losing.
We were afraid [that] it wouldn't be successful.