Among other things, this method of taxation tightly couples school funding to local home values. Areas with expensive homes have the best schools. (And areas with cheap homes have bad schools, which is not really what you want if you are trying to do something about inter-generational poverty.) This is self-reinforcing because now parents compete with each other to buy homes that are located in the best school districts, which drives up prices even more.
When home values drop, such as they did in 2008, property tax revenue drops too (Prop 13 is not a one-way ratchet, you can get re-appraised and lower your taxes, and 2008-2009 was a pretty convenient time to do that).
It doesn't seem like a good idea to base city revenue on something that is so undependable, especially something that is so often treated as a speculative investment.
What I would rather do is come up with a system where a simple tax is paid per parcel, perhaps a flat tax or one based on the size of the parcel, and which is not based on the market value of the parcel.
The rest of city funding can be covered by income tax -- unlike the current system, at least that has a chance of being progressive. No old people will be priced out of their homes by tax increases, and school funding would be distributed somewhat more evenly because not all high earners live in the most expensive areas. And if you are worried about an influx of high earners flooding into your area and driving up the prices, a local income tax would certainly discourage them.
