There was a study which showed closing the lid reduces the acute problem but actually increases dwell time.

https://www.researchgate.net/publication/339650907_Real-time...

The larger conclusion is that the health consequences of this alleged horror seem to be, in fact, fuck-all

Well, as long as the closed toilet bowl has time in between to settle, it's all fine. And your hotel toilet isn't exactly a high traffic area.

Generally you would need everyone else to do that for it to help you, which isn't something you control.

Doors are nice from the public health perspective in that people actually do usually close them without even being asked.