That sill doesn't answer the question of why God proving his existence to a non-believer would take away his or her free will.
Because many non-believers, deeply within their hearts, do NOT want to learn about God and the way of love and everything because they will feel forced to obey (though God does not enforces anyone to do so) and they really do not want to. They love their hedonistic lives, their hatred, their sins and they do not want to change. For example, how many guys or girls today REALLY wish to remain virgins till their marriage (I've chosen this example because it is a very common one). In situations like that, God is better to let these people the way they are and not prove His existence to them, otherwise it will just make things worse.