re: What establishment(s) provides the best public restroom?

Being a man, most "women's stores" have pretty clean restrooms (teacher supply stores, Hobby Lobby, Ann Taylor Loft, etc).

The restrooms are hotel lobbies are generally pretty clean IF they don't have a lobby bar. While traveling I frequently drop a morning deuce at a Hampton Inn's lobby restroom (and grab a free banana or apple).

For chains, Tractor Supply Company surprisingly has really clean restrooms along with Walgreens. Also Lifeway bookstores are generally real clean.