still pays the same dollar amount tax that they did in prior years. If a bubble bursts and housing prices all shrink to to half, then each property owner still pays the same amount.

The taxing authorities specify a total amount to collect. The county assessor estimates a appreciated value for each property. Then each property pays it's proportional share of the total.

In practice, some properties appreciate faster than others. That can cause a taxes to go up a bit on a particular parcel even if the taxing authority does not collect any more tax. There's also a carve-out of the first $100,000-$200-000 for private residences. But a doubling of housing prices will never result in a doubling of property tax unless the taxing authority also decides they need to double the amount that they need to raise.

If Prop 13 saved people from losing their homes when housing prices appreciated over the years, it's only because other parts of the tax code allowed taxing entities to increase the tax amount.