Why hold the rim? Right way up or not, the part that's being touched goes directly onto the glans (where the mucous membranes are, and infection occurs). If you're worried about infections spread by touch, your method offers the same outcome as putting fingers inside the condom, because it is the same. To avoid this, the tip, specifically, is what's held because once it's rolled on, the clean inside of the condom is the only thing contacting the penis.

If the tip is held on the wrong side and applied, you throw it out and grab a new one (for her protection). Also, the tip needs to be pinched so any air is removed.