Absolutely NOT true.
Med school adcomms consider the name on the diploma to be of only minor importance when it comes to making admission decisions.I
Annual survey of med school adcomms–
See p. 14 https://www.aamc.org/media/18901/download
UCLA is the largest supplier of med school applicants in the country; UCB is the fifth largest.
https://www.aamc.org/media/9636/download
Either school provides the opportunities you’ll need to make a strong med school application–developing a strong application portfolio is entirely up to you.
Take pre-med out of the calculation and pick the school that offers you the best fit and cost.
BTW,@aunt_bea is right. CA public medical schools (except for UCR which prioritizes residents of the IE region) do not offer a strong in-state preference in admissions.
CA is one of the most difficult states for med school admissions. Only 17% of CA applicants matriculate to an in-state (public or private) medical school.
https://www.aamc.org/media/6016/download