No, that takes way too much time. If you check using a UV light you’ll notice most simple forgeries but I guess most people just don’t expect small bills to be forged.

Does it? Every time somebody pays in cash in a store here they run each bill through a machine that, I'm pretty sure, checks the authenticity of the bill.

A store is not a bar. Stores typically have less transactions but these transactions involve more money, so you can afford more time.