Our coach forbids (us) 'drinking'/'to drink'
Solution 1:
The reason is that the verb forbid is transitive but not doubly transitive. That is, it needs one object, but cannot take two objects. The object, however, can be either the thing forbidden or the person to which it is forbidden.
I forbid it.
I forbid you.
(You can't just say "I forbid"; you need the "it" or "you", but either one works).
Our coach forbids alcohol to the team.
Our coach forbids the team to drink alcohol.
I believe some dialects permit a doubly transitive forbid, but most dialects don't:
*Our coach forbids the team alcohol.
Those people who do permit it would allow your last example.
This happens with other verbs as well, like provide, which is only doubly transitive in America—see this question.
Solution 2:
"Forbids drinking" is OK because the gerund (V+ing) is the object of your verb.
When you want to mention both the person and the act, you can choose between:
forbid somebody to V
forbid somebody from V+ing
You can say "Our coach forbids drinking," and "Our coach forbids us from drinking."
Personally, I prefer to use "forbid to," though.
Solution 3:
Perhaps it could be said that Our coach forbids the team alcohol contains an implied to, making the team an indirect object; i.e. Our coach forbids to the team alcohol. (Admittedly ...alcohol to the team sits better).
Similar to Give it me! = Give it to me!