Of course, other than perhaps the gross factor - I'm not sure how dangerous it is brushing your teeth with remnants of fecal matter on it. So many people leave the lid open yet the hospitals aren't full of people with Hepatitis A.
All odor is particulate. Think about that. Does a toilet lid contain odor? No.
We rebuilt our bathroom 15 or so years ago and put in a closet for towels and bathroom stuff in general. In the closet, on the wall is the toothbrush rack right where I like it.
Im sure this will be ‘too much information ‘. But As an older guy I have to urinate every 2 hours overnight I run in.... Pee.....then get right back to bed. The seat and lid stays up. And I rarely flush in the middle of the night with just liquid in the porcelain convenience. Life’s a beach. Whatta you gonna do?