If you aren't sure that you'd abort the fetus if a genetic abnormality were detected, you'd probably be less likely to take a screening test that poses even a minor risk to the fetus. (of course, there are other reasons to know, like if you are considering a home birth versus hospital birth, you'd want to know if the baby will have any special needs at birth)
I don't know a lot of people that refuse the noninvasive prenatal screening offered by California... even if you don't act on the information you receive.
Back from when I was pregnant, I also recall that my insurer (Kaiser) would cover the cost of NIPT for anyone 35+ or in certain high-risk cases without having to go through the state. They also covered the cost of carrier testing.
edit: I thought maybe you might be thinking of newborn screening which is also very much a Thing in CA, but it turns out that is also not free - $130, also covered by most insurance. Good to have though!