27% of Californians are over 55. That means the majority do not get to have the benefit of keeping their tax bill when they move. To compound that, the most dynamic economic growth probably comes from under-55s moving, not over-55s -- new jobs, new opportunities, start a family, etc. Obviously it's good if it's easy for old people to move, but the economic value of a young family being able to move closer to a high paying job is probably many multiples more from a "net good for society" standpoint.
Prop 13 freezes your tax rate when you purchase the property. So the subsidy isn't given only to those who buy before the bill was passed -- it is given to anyone who buys before housing goes up, at any time while the law is in place.
It's extremely common for people to borrow against the equity of their home (even moreso if you are very wealthy). I think the parent is arguing that those who own property are borrowing against the value of their (subsidized) asset. Another thing to consider is that a yearly tax on an asset can significantly affect the valuation of that asset, far in excess of the amount of the tax itself.
Of course there's many reasons for the embarrassing lack of ability to solve the housing problem. Prop 13 is a big reason, certainly one worth looking at when considering solutions to this problem.
It's also just unfair. Most people who are subsidized by prop 13 are subsidized for no reason at all, just market timing. And there's another layer to it, which is that the earlier you have bought, the more you are subsidized. Guess what was more difficult for minorities to do 30 years ago? That introduces an additional dimension of racial unfairness to this. Not to mention, it's a subsidy for the rich (homeowners) paid for by the poor (everyone else).
Prop 13 is an anchor around our neck. It's kind of like that excruciatingly painful disease that causes your bones to fuse together, except for growth and prosperity in the state of California.