Interestingly there is no word in the manual and nothing to be found on the internet. But it seems like an obvious question:
While a UPS (let’s use APC Smart-UPS 1500RM as an example) drives equipment/server, is it safe to disconnect the mains power cable?
Which implications does this action have?
I have done this in the past and did not have issues. However, it does not feel right because that would interrupt earth connection as well. How come then, that the manual contains lots of obvious warnings but does not explicitly warn against disconnecting during operation?
Assuming everything in your electrical is properly connected, there is very little risk to doing this. However, as you mention, if there's a short anywhere along the electrical path you risk becoming the earth/ground connection. It's the quickest way to test, and I've done it many times myself, but it's probably not the best way.
It's a valid comment for you to point out the oversight.