あまりに(も) strengthens the degree of exceeding, but probably it does not make much difference. One important property of this kind of adverbs is to compensate the head-finalness of Japanese. In Japanese, the head of a phrase comes at the end of it. Therefore, as is often pointed out, you cannot tell whether a clause is going to be negated or not, interrogative or not, etc. until you reach the end of the clause. You cannot position the head in the front, but adverbs can be positioned in the front, and if a particular adverb shows concord with a particular head, then that will give you a clue before you reach the end of a clause. In case of あまりに, it often matches すぎる (but not necessary require the existence of it)
so the moment あまりに is pronounced, you can tell that the clause will have the meaning of exceeding. Similarly けっして concords with negation, and はたして with interrogative.
They function as giving clues to what type of sentence it is going to be before the end is reached.