The reason for Prop.13 was to prevent the seniors from getting taxed out of their homes. They can take their tax basis with them if they decide to move, so there is some portability.
If anything, fixing the tax rate effectively encouraged higher pricing because a buyer didn’t have to worry about further gains beyond their purchase price increasing their cost of ownership.
It does limit the tax revenues to the state and local governments, though, and is a leading cause of the increases in fees, bonds, and development hurdles the cities place in front of the developers.
The annual tax bill for a $500,000 at 1.5% would amount to $7,500, or $625/month. If the property values doubled over a 3 year period as they recently did, EVERYONE’s property tax bills would likewise double. Obviously, that would cut into refinancing options for a lot of borrowers, and all buyers would have to consider the risks of those increases when they bought.