You're missing the point of removing the dedicated structure. They are public spaces even if they're private property because they're open to the public. I could right now look up a local church and go visit it. By removing the public spaces it becomes much harder for the religious leadership to shape and control narratives.

A significant reason why religion in the US has become so problematic is that the religious structures are easy to find and easy for large groups to regularly attend at and therefore make very convenient locations to distribute propaganda from. By removing the central structure you force things into a distributed system that's much harder to weaponize. You also eliminate nearly all of the power of the religious leaders as without the regular services their authority is diminished to almost nothing.

As for the criteria being vague, no it really isn't. You just define it as religious and there is already ample legal precedent on exactly that topic. There have been laws, regulations, and rules about religion in the US pretty much since the founding of the country, mostly in an attempt to prevent exactly what's been happening lately which is Christianity being written into US law.

It doesn't have to be perfect, people will definitely find loopholes, it just needs to make it hard enough that most people won't bother. Religion is already dying out, it just needs a little push. Remove the tax exempt status, remove the dedicated religious structures, and the rest will take care of itself. When people have to make an effort to attend religious gatherings, most won't bother.