I've heard that this kind of s(h)itting is (or was?) common in some Asian countries. Learned that when someone left shit all over the toilet at our company, in places where it would be basically impossible to get to unless you were sitting like this.
Many Asian countries used to have squatting toilets (if any). Those are becoming less common but I guess you'll still find people who have never seen a western style toilet. This kind of sign is not at all common in Thai toilets so there must have been a specific incident that caused them to put those up. They also weren't there when I was in the same place last year.
I still do it wrong sometimes if it's going to be a hard poop, but I lift the seat and keep my shoes on. I don't see how this could make a mess unless someone was heavy enough to break the toilet.
The squat makes it easier to poop, but there are modern toilets that look similar to Western seat toilets, where if you sit on them, you sort of naturally squat (but not on the ground, but raised). That should be a good improvement.
Or, you put up something in front of the toilet, like one of those tiny sits, that you put your feet on.