Adam Smith himself argued that the portion of property tax derived from the land value cannot easily be passed on to the tenant. That portion is high in some important cases (detached houses, warehouses, and factory buildings in outer suburbs).

The reason is that rents compete on net profitability to the renter, not revenue. If rents go up, tenant profit goes down, and they move to lower value land further out where the lower revenue minus lower costs gives them their original profit. Buildings are affected uniformly by property tax and cannot be escaped in this way.

Therefore, your prediction would be most accurate in cases where building prices dominate the optimal land usage. Non-optimal uses would get redeveloped, as you point out, and building tax can be pushed to tenant, but the land tax component must be eaten by the landlord. The exception is for properties in a market where the owner occupier distortion allows them to sell at a higher price (can sell to a renter now or could redevelop for owner occupier in the future).

No. The suppliers (landlords) as a whole have always been charging the maximum that the market is able to bear. Just because rents are raised everywhere doesn't mean the market can bear it. It would push every renter to cheaper smaller worse units and push the poorest renters to homelessness.