He doesn’t need to specifically address reincarnation for us to know it isn’t a possibility. The failure in your logic is trying to assume if something isn’t specifically addressed and denied, than it might be true. It’s like asking why won’t God make pizza rain showers because the gospel never says we can’t so maybe we can. Jesus tells us what will happen after we die and tells us how to follow Him. All of the gospels speak as if talking about one life. The only afterlife we can look forward to is the one with God. I think the major difference between Christians and many other religions is that our love for God and desire to be with Him after death drives us. If we start desiring more time on earth than allotted, whether it’s by searching for a ‘fountain of youth’ or hoping for another life, we are failing to love God enough to desire heaven. Or, we fear His judgment and wish to put it off. Both traits separate us from God. Reincarnation isn’t possible for Christians, not only because it isn’t in the gospels, but because to desire such a thing goes against what it means to be Christian and to desire to be in God’s presence. Read the gospels for yourself and you will find that this is not a possibility for Christians, if you truly understand what it means to be Christian.
The second part of your response is a very odd, hypothetical question. We follow Jesus Christ. Period. He has told us enough about His second coming that I don’t think anyone is going to be asking to stick around on earth instead of going with Him. He gave us the Holy Spirit to guide His Church on earth and, so far, the rules haven’t changed. If the Holy Spirit guides the Church in a different direction, than we will follow. I don’t expect they will because it would go against everything he has taught us so far (and against our love of God and desire for God).
Additionally, you must understand, not just the gospels, which are very important, but also the Old Testament, which predicted the events in the gospel. Jesus did not do anything that was unexpected, if you read what the prophets wrote. The suggestion that He would completely reverse His own teachings and the entire foundation on which His teachings were built on seems not only improbable, but ridiculous.