Does Medicaid pay for an assisted living facility?

Actually in your state, California, and in some other states, Medicaid/medi-cal DOES pay for assisted living. Including room and board.

Many of the states that do pay for AL do so in a very limited way.  Not every facility accepts Medical, and there are limited spots

You would need to apply for the Medi-Cal Assisted Living Waiver program. There are a limited number of spots available, and only in certain California counties. Eligibility is based on needing a nursing home level of care. The waiting list is long, currently around 12-18 months, unless the applicant is already in a nursing home. In that case I believe there is a separate and shorter waiting list. You start the process by contacting a Care Coordination Agency to get on the waiting list.

There's more information and a list of Care Coordination Agencies here:
https://www.dhcs.ca.gov/services/ltc/Pages/Assisted-Living-Waiver Program---Medi-Cal Member Enrollment.aspx