That's largely true, though tax law does reduce the tax hit for people over 55. California recently amended prop 13 so that people over 55 can carry their existing property tax 3 times, and modified it so that they can purchase a more expensive house and keep the lower tax burden on the amount they got for their previous house. They also greatly reduced the extent to which heirs can inherit the low previous property tax (I think it's now limited to 1,000,000 and has to be a primary residence).
The inherited low property taxes always seemed pretty egregious to me (almost like an aristocracy of landed wealth). The whole thing is pretty dismal, honestly - I'm not opposed to low property taxes, but a tax regime where people pay vastly different taxes on similar properties causes all kinds of problems (and is fundamentally unfair, in my opinion).
Much of the change to the law, though, was driven by real estate agents who want to increase transactions. The previous law preserved low property taxes as long as elderly people didn't sell and left the house to an heir. Now, people 55 and and over can sell repeatedly without increasing their taxes, and there's more incentive for heirs to sell the house. That's as much was 4 commissions where there used to be none.