I can’t claim to know the benefits of state, local, or fed taxes. Like a lot of things, I imagine it’s better trialed on a local then state level, and might never reach federal.
Like a lot of tax claims, it may just be reliant on claims, and would not always require proof. You file for taxes, you report 3 homes, you state which one you occupied; or you state that you had a tenant in that home. If you’re audited on your taxes, they may find you falsely reported a tenant, which would be tax fraud. The IRS could find reason to audit someone if, for instance, they’re freely posting on Facebook “Yeah, just say you have tenants who do not wish to be named, they can’t do anything about it”. Many tax rules already work by self-reporting, and/or finding conflicts in prior documentation.
It would not prohibit third homes, but you’d have to pay a hefty property tax to hold onto an extra home and require it to stay empty while people are out there homeless. So, you’d have to be rich and not care that simply owning these properties bleeds you money (which is the opposite of how being rich usually works - your properties generate so much value by “existing” that you can simply persist a high quality of life just off residual income)