You're point about Christianity being spread only through man does not account for various things,
such as
a) The Anonymous Christian, those who don't know it may still be under the guidance and grace of the Holy Spirit, this isn't some wishy washy new age belief either it's in the Catholic Catechism
1260 "Since Christ died for all, and since all men are in fact called to one and the same destiny, which is divine, we must hold that the Holy Spirit offers to all the possibility of being made partakers, in a way known to God, of the Paschal mystery." Every man who is ignorant of the Gospel of Christ and of his Church, but seeks the truth and does the will of God in accordance with his understanding of it, can be saved. It may be supposed that such persons would have desired Baptism explicitly if they had known its necessity.
b) That God can work through people, You either doubt God's power or His sovereignty in saying the Church only spreads through men. If we ignore dramatic miracle conversions (like St. Paul one of the greatest evangelists) there still is no logical reason for God not to choose to spread his Church only through His people, because it pleases him to do so, you can not judge God.
c) A Possibility of Predestination, for all we know God could have choose those who He will save and those who he will not, and it may be due to a grace irresistible, in so much as the elect will recieve the Gospel one way or another.