It is certainly true that highly ranked public high schools can be insanely competitive. I have for example heard stories from both daughters of students freaking out because they only got a 97 on a test. We also see posts here about students who are stressed out about AP classes, ECs, and many other things. The number of high school students in the US being treated for stress related illnesses is very high and to me seems quite unfortunate.
One thing that concerns me: The top ranked universities are also very stressful. I do not think that MIT or Caltech are any less stressful than the most stressful high school in the US. You really need to want to do it for a school at this level to be appropriate. I think that this concern applies to any “top 20” university.
There are a lot of very good universities in the US. You can get a very good education at a wide range of universities and colleges. Of course the pubic university system in California has a lot of very good schools (and not just the top 2 of them).
It is hard to give a definite answer without knowing a lot more. However, I am thinking that you are best off attending a high school that is a good fit for you, and where you can be comfortable and where you can do well without being stressed out. Then you are best off looking for a university that is a good fit for you. You will be a few years older when you are applying to and then attending university. However, I would not expect UCLA or UC Berkeley to be any less stressful than your current high school.
I am also guessing that the word “worse” is not really quite right. Perhaps “less exclusive” or “lower ranked” might be a better wording.
I do have some experience at attending highly ranked universities. When I was an undergraduate student this was not a good fit for me. When I was a graduate student I was older, more mature, and loved it. The same person can be more ready or less ready to take on the stress at different points in their life.