But you'd sit on the rim, if the seat was up. I doubt there are any people (women or otherwise) who would stick their arses that far into the pan.
The bowl is big with the seat up. Plenty of room for an unsuspecting ass to fall right in. Why do you think it bothers women?
Psychological thing and comfort. The rim of the bowl is actually no bigger with the seat up than with the seat down. It's just narrower without the seat.
BTW, what if you don't have alid that closes over the seat?
If your dunny is in the bathroom, which is commonplace in the US hence the term 'I'm going to the bathroom', you can always wash off any water that spashes up from the bowl. But seeing as Adam & Jamie busted the germs myth, the point becomes moot.