I think (guessing a bit here) is that the argument goes like this: repealing Prop 13 would drive down house prices, allowing some renters to instead buy/mortgage, therefore reducing the demand for rentals, driving rents down.
By increasing supply, I would imagine, at the expense of older, retired home owners. In my community, there are a lot of old people that wouldn't be able to afford a fraction of my property tax. Many would be forced to sell and there would be a glut on the market of new homes, which would likely drive prices down.
I hate paying the property tax rate I do, but I would also hate to drive all the old people of my community to cheap, depressed housing stock in some remote place.
What's the way out of this? I really don't know.
Prop 13 as it stands is simply a way to allow commercial landlords to avoid paying property tax at market rates.
It would make sense for governments to formalize this arrangement and allow deferring of property tax payments until the house is sold.