The draft has nothing to do with monopoly exemptions (which only talks about broadcasting rights: https://en.wikipedia.org/wiki/Sports_Broadcasting_Act_of_1961) and is purely a function of the structure of the business. The NFL is a hub and franchise business, there is a core structure and the teams are franchises that follow rules set forth by the hub. It’s no different than McDonalds, which also has rules about grabbing employees from other franchises and sets salary structures. A new league would need no such exemption, and indeed neither the XFL nor USFL did. It’s all about how the business is structured.
You are correct in that it how the business is structured. Pro sports could look very different but still exist.
The draft is a violation of the Sherman Antitrust Act, but is made legal by the collective bargaining agreement with the players union.