Mythusters found fecal particles everywhere even when the lid was down. There’s no escape from gross, just finding an acceptable level.

Then all the bacteria are flung onto the lid, and when I sit down I might touch the lid with my back.

There’s plenty of bacteria in the air already, a million more aren’t going to hurt. Unless you have a norovirus, then you need to disinfect with bleach after you’re done.