This depends vastly on your interpretation of who is allowed into heaven or not. If belief in Jesus alone is the only means of entry, then yeah, your fate depends on luck (whether you're ever exposed to Christianity) in spite of your free will, yet under the system where those who lead a Christian life whatever they actually profess gain entry into heaven would validate free will.
Either way, it's not a fundamental flaw of "Christianity" as a whole. If i could be so bold as to make a claim, the coming of the Holy Spirit through Pentecost was the instance that gave the whole of humanity the capacity to be Christian (whether or not they'd received the word, they had received the spirit), so it fixes that problem.
(note that i'm an agnostic with a moral system that isn't spiritually-grounded).
Monster Hunter: pissing me off since 2010.