It does have the problem you describe of landed gentry not getting hit with high property taxes, but also it does protect middle class people from getting kicked out of their homes due to the popularity of the area they live in. My parents are the perfect example of this, my dad is a plumber and my mom is a teacher, my entire life they barely broke 100k combined and were BARELY able to give me and my sibling a decent life in LA. If their property taxes had risen with the rate of the area that they bought a house in as we grew up, we would have been unable to continue living there and forced to leave. The same thing goes with my grandparents.
California has very high income tax, which arguably compensates for the low property taxes. This strikes me as more fair, as the high earners are the ones hit with the tax bills rather than my grandma, who made nothing and had no way to pay the tax that comes with a $2M property valuation, despite having worked hard to buy her house when she was younger.
So yeah, you're right in some sense, but its not fair to completely ignore the alternative of lower/middle class people being kicked out of their homes if they didn't have prop 13. The real problem here is the NIMBYism that you describe and restrictive zoning.
ALSO, the landed gentry are sort of limited as well. Yes their primary home (the one they live in) has a limited tax bill, but additional homes they own, they will probably end up renting out, and that rental income will be subject to California's high income tax.