Yes, of course, we are not barbarians.

(Just to be clear: I think doors on bathrooms are great. I just don't think they really contribute to hygiene or health--apart from mental health.)

Airborne fecal contamination from flushing is likely reduced by having a barrier

Like a toilet lid?

It helps. Know what also helps more? A second barrier after that one.

I’m sure a third would also help even more (front door of the house?) but most people consider that a bit excessive for daily use within their own domicile.