My friend in Minneapolis brings her cat to the coffee shop. She does it the same way you do, with a leash and a carrier as a backup in case the cat wants to hide.
After a few times, the cat was very comfortable at the coffee shop, and the total star of the show!
Yeah I am happy I can take my cat anywhere too.
You can also find out which people suck, because if you don't like cats, you probably suck.
Red flag for me !