The forcing no trump response is sort of a hack. The key idea of the 2/1 game forcing system is that a non-jump new suit response at the two level to an opening one bid is forcing to game. In order to make that work, you have to take some of the hands that Standard American puts into 2/1 responses (specifically the non-game-forcing hands) and do something else with them. The 2/1 system chooses to put them into a 1NT response.
Why would you do that? Basically, 2/1's main advantage over Standard American is that it has improved accuracy in slam bidding. Because responder can often force to game at a low level, nobody needs to jump later in the auction with only moderate extra strength. This gives more room for slam exploration. The main disadvantage is decreased accuracy in partscore bidding, especially with regard to being able to stop in 1NT. Many people find that this is a good tradeoff at teams, and fairly neutral at pairs, so they play 2/1 instead of SA.
As to the "three point range" aspect of your question: All else being equal, it is good for one hand in the partnership to be able to show a balanced hand in a narrow range. That is why no trump openings and rebids by opener tend to show a balanced hand with a 2-3 point range. On the other hand, a 1NT response to an opening bid doesn't even promise a balanced hand (even in Standard American's nonforcing 1NT response). The forcing NT response asks opener to further describe their hand, rather than showing a specific strength or shape. While neither partner is yet captain, responder knows more about opener's hand than vice versa, so opener should describe further while responder looks for sufficient information to take full control of the auction.