Even in the US? Because as the link the grandparent comment shows, it is one of the words Google Docs is warning away from as insufficiently inclusive

Yes, even in the US. But of course, given enough influence, you can make absolutely anything non-inclusive. And social pressure will make everyone have to accept it.

I also find it baffling that inclusivity folks find a term that used to be male exclusive becoming applicable to both male and female as problematic. Wouldn't they want something like this to happen?

I don't see this as too baffling- in their logic, if you have two alternatives (landlord and landlady, say) and landlord gets picked, it's evidence of patriarchy (because it defaulted to male). They would prefer "landperson", or perhaps "rent seeking capitalist"

Yes, just like "actor" is gender-neutral (although "actress" is still explicitly female).

When I rented a room from a woman in Atlanta, the other tenants and I always referred to her as "landlord" and nobody batted an eye. The only time I hear "landlady" is when One Bourbon, One Scotch, One Beer plays on the radio. So that's my datum.