Even Christ talked of reincarnation, though he didn't use that word.
I've remembered thousands of past lives, mostly only as glimpses. But I've had a few profound memories overwhelm me.
Let's start with the basics: we are basically immortal spirit wrapped in Homo sapiens flesh. (Genesis 1:26 + Gen 2:7; also see Gen 6:3).
We fell from grace, fell into our own egos, blinding ourselves from the spiritual world, so that we became dependent upon physical processes (like human eyes) in order to perceive the world around us. In moments of spiritual clarity, I have seen without these human eyes.
That great fall from grace required a rescue mission. That's why God created Homo sapiens -- the perfect form for temporary vessels of consciousness. God wants us to have every possible opportunity to escape, thus reincarnation. Look at Numbers 14:18 from the viewpoint of reincarnation. The sins of the fathers are visited only on the guilty! Look at that. Not innocent children. It's all there.
As far as the Solar system ceasing to exist, that won't happen for likely trillions of years. The Earth, however, will likely be destroyed in about 5 billion. But life on Earth will be wiped out far sooner -- likely 1-2 billion. God will likely be leaving this world (as far as the rescue mission goes) in only a few years -- likely a thousand or less. In the thousand years of peace after Armageddon, we will have the choice to return to God or to stay. But those who stay won't enjoy the fruits of civilization. That bright and shining creation only came with the wisdom of God and religion (spirituality).
Those left behind will resort to the physical laws of action-reaction -- wailing and gnashing (victim-perpetrator). Reincarnation will continue as long as there are bodies to be "worn." But the rescue mission will be gone. And when Earth is gone, the spirits might drift across the universe finding some other planet with life. Spending eternity as an amoeba or some furry or scaly creature without intelligent thought doesn't appeal to me.
I personally hope we all make it to spiritual awakening. I've tasted that "wine" and it's the best of all possible things.