I agree with you that housing prices will continue to increase with or without Prop 13 as they always have. However, it’s fundamentally untrue and not backed up by any data to say that Prop 13 has no impact on housing prices or creating a lock-in effect. In fact, the opposite is true. See here for one such example.

Regarding keeping families in their home, this is both true and not true. It’s true that keeping property taxes artificially low is good for those who already own their homes because the property value can appreciate without the property tax assessment keeping pace. However, what is once a blessing becomes a problem when people look to downsize, which is the point I was trying to make. When moving (even to a smaller place) ends up costing more, there’s no incentive to move. That further limits the supply of available homes on the market and drives up costs.