I agree with your point there, particularly that cities will look for regressive sales tax rather than property tax.
But the effective elimination of property tax on commercial real estate vs. residential real estate also has the problem of hugely distorting the market. When I was looking for office space for my startup around ~2010, there was enough empty commercial space to for every resident to have 200 sq. ft.!
Meanwhile, residential housing stock is so low that the rental market is largely driven by personal acquaintances. I remember calling numbers for 30 ads, and only getting a response from one, because they were like "hey I know that guy!"
This is also a zoning issue. But commercial real estate acts like an investment, and if empty commercial real estate is profitable to hold on to because rents are low and taxes are low, it makes it impossible for a market to properly switch between the two, even when all that commercial real estate can be zoned as acceptable for residential use.
In any case, I see no reason why commercial real estate should be taxed less than residential real estate, unless it would be desirable to distort the market in that way. (And depending on other forces it definitely could be. I just know that Prop 13 is a really bad deal for my county's economy, and don't know why the State should have the power to override the county or city on this.)