They wouldn't be homeless if they already owned property. Even before Proposition 13 the median house price-median income ratio was higher in California than elsewhere, around 4 while the rest of the country was at 3. So they could do what many people do in old age, they sell their house for a large sum and move somewhere cheaper.

It was true that before Proposition 13 many older homeowners who could not keep up with rising taxes were forced to sell. But taxes were rising because house prices were rising, so they were able to sell at a nice gain.

Why would old people be homeless? The cause of them being forced out is that they own an immensely valuable asset but live on relatively little income. That asset would then presumably be sold for cash. Any house expensive enough to create this situation would also give them enough hard cash to live out the rest of their lives in relative comfort.

This is essentially a relatively benign form of eminent domain, which has the same consequences without the implication of a good financial outcome.