Nobody can know that reincarnation exists!
That is not correct. You may speak for yourself. But not for everybody.
To say that something does not exist simply because we do not have the experience or proof is an example of wrong work of the intellectual center. That is the sin of skeptics. If we do not know we simply acknowledge we do not know. That keeps the mind open and then we may learn new things. In this case, learn that the manifestation is much, much richer than our limited physical reality demonstrates.
There are countless people who have metaphysical experiences and direct knowledge. This does not in any way contradict the Realization of our True Nature and, while metaphysics can become an obstruction in certain cases, in other cases it can actually help.
How can one know it exists?
Intuitive knowledge, memory, and awareness of the depth and age of the soul. Until that is established in direct experience there are techniques that can be used to educate our perception, should we be interested. Most people are fine without that knowledge and such knowledge is not a condition for Spiritual Realization. I have met people who appear to me to be sages who have no metaphysical knowledge at all.
You know how old you are in this life, and you have an awareness and knowledge of the things you experienced and the changes your personality, character and body have undergone in this lifetime. Likewise, there is a subtler aspect of our manifestation as individuals that has the same perspective of experience through multiple lives. This personality and this name die with this body, but a subtler “person” persists.
The end of the cycle of incarnations has no causal relation to Spiritual Realization or vice versa. The realization of our True Nature is independent and unrelated to those experiences; although maturity and wisdom gained though time certainly has an impact on it happening in this lifetime or not.
When we talk about the “ripeness of the seeker” that is exactly what we are talking about: soul maturity. It is the soul that is ripe, not merely this passing personality. It is a much deeper, broader and substantial part of our being that invites and is “ripe” for Realization.
We just need to be a little observant to notice that different people have radically different soul ages. There is no time in one lifetime to grow from the most ignorantly behaving person to the wisest sage. If we can set aside our overly intellectual compass for a moment we will notice that this can be taken as evidence that our lives are much, much longer and richer than only this lifetime.
It takes more than one lifetime on the physical plane for all of the nonsense we have learned to take for truth to be eliminated from the psyche.
The focus of spiritual work is this lifetime alone, certainly, or to be more specific, this moment alone. How could it be otherwise? This is it. But the notion that only this one lifetime is available to allow us to go from immaturity and stupidity all the way to Love, Truth and Beauty is not realistic.