What public places are barefoot friendly?

All or most Airports (but usually not the flights). Restaurants, grocery stores, ect. will usually not say anything at all, even if they have a sign (most don't). They know that there is no legal basis to exclude a barefoot person. Only the most pretentious of public places will confront a person.

The trick is to look presentable in other ways and keep your feet looking clean. Be confident. Wearing long pants can help. Remember in the US you have a right to public accommodation without discrimination.

Life, liberty, and the pursuit of happiness. If you don't like it, move to North Korea.