Many vacancy taxes already exist all around the world. There is not a single one that taxes normal short vacancies. It is just false that this increases costs for all landlords. The vast VAST majority of landlords will never pay it.
On the other hand, the increase in supply due to the tax can be noticeable, which has a much bigger effect lowering prices.
If the landlord can increase rent by $100 and the market will bear that, why is the lack of a vacancy tax stopping them? Landlords charge the maximum that the market can bear.
It’s true that inflation presents an arbitrage opportunity, but I seriously doubt the average consumer is thinking with anything close to that level of sophistication. I mean, most consumers leave debt sitting on their credit card instead of paying it off when they can.