The basic reason no self or soul is reborn is neither exists independently of the mental universe that gave rise to our illusion of selfhood.
The mental universe within which we all exist is dynamic and so are we. In Buddhist terms, this dynamism is action or karma.
Buddhism does not say we do not exists. It only says that our selves are empty, that they do not ultimately exist. When we die our karma, the mental activity of this life, reconstitutes as a new being ensconced within the larger mental universe.
No one explains this better in modern terms than Bernardo Kastrup. In his essay Making Sense of the Mental Universe, he does not write about rebirth but rather about the conditions of our existence within the mental universe.
Nonetheless, his explanation of a “mental universe” shows precisely how rebirth can occur without there being any soul or pudgala or anything else that flies from the body upon death to transmigrate to another one.
I highly recommend reading the essay linked above. I have no idea if Kastrup is a Buddhist thinker. It’s even better if he is not, if his thinking arrived independently at a place consonant with original Buddhist thought.
Most Buddhists know that even Buddhists have trouble understanding how someone can be reborn without having a soul, self, or pudgala. What did the Buddha even mean by that? I know more than one university professor of Buddhist studies who explains Buddhist rebirth by saying, there is no such thing and neither is there such a thing as karma.
Those professors explain away karma and rebirth by claiming those fundamentals of Buddhist thought are nothing more than the Buddha “using the concepts of his day” to teach his moral doctrines and what amounts to his “atheistic Stoic” philosophy.
I mean no disrespect for the professors. It is hard to understand how something can be reborn and yet be empty of any perduring self or soul.
The essay linked above provides an excellent explanation of how that happens. I strongly encourage Buddhists or people who teach Buddhism or are interested in it to read Kastrup’s essay when you are in a good mood and want to learn something new and really interesting.
This essay can give you another angle on Kastrup’s thinking: Matter is nothing more than the extrinsic appearance of inner experience.
And here are some of my comments on Kastrup’s essay Making Sense of the Mental Universe.