The motivation is really uncompelling. It describes what intersections do, but not why we need them. I think it would be better to dig through the mypy issues to find various problems that require intersection to solve them and present them under motivation.

For example, this one: python/mypy#9424:

from typing import Optional, TypeVar

U = TypeVar('U', bound=Optional[int])


def f(u: U) -> U:
    if u is None:
        return None
    assert isinstance(u, int)  # removing this line causes it to pass.
    return u

works with working intersections since u will have static type int & U.