As a person who has been a person of faith, no faith, and is now somewhere in between (believe in a God, but beyond that I run into roadblocks), I would like to take a stab (again, for those who do not know, I am not yet a mason, but will petition once I am 21) at this question.
I think it is important to recognize that public masonry (via 1717) came out of a specific culture, time, and place. Within the context of a Christian kingdom, few people were publicly atheistic. Public atheism is relatively novel. This context provided, I think, a primary reason for this requirement.
Second, tradition. Most of the Masonic documents available reference a Supreme Being/God either explicitly or implicitly, and when traditions are broken, fractions occur (GODF split, for example). Reactions to fractions have many forms, but most prevalent is the tendency for groups to harden their stance on issues that cause a rift. After GODF split, mainstream masonry defined itself by its fidelity to traditions/landmarks.
Third, there is the stated reason you mentioned. The same reason in courtrooms people swear on the Bible is it serves, so the logic goes, as an impartial assurance of honesty (yet that honesty only depends on the integrity of the person).
I don't think a man needs religion to be a good person, and I don't think a belief in God, however sincere, makes one person inherently "better" than another. I think Masonry requires belief because, with the above, there is an older (and IMHO, erroneous) understanding that the atheistic person cannot be moral, which is a steep indictment for a fraternity that prizes moral uprightness.
Okay, I've said my piece.