Now the robots have been banished, just to give your thread a bump for others, I feel I should add something semi-useful.
Not a pub, but somewhere I've never seen mentioned on here is City Cafe. I used to be in there all the time when I was around Edinburgh regularly 100 years ago. I've Googled and it's very much still going and looks almost identical. Good diner food and drinks. Probably not what you're after coming from the US [edit: eek! Canada! My apologies.] :) It was always a very cool crowd back in my day, and I get the feeling it's still not really on the tourist trail even though it's in the Old Town.
https://www.tripadvisor.co.uk/Restaurant_Review-g186525-d720716-Reviews-or15-The_City_Cafe-Edinburgh_Scotland.html
https://maps.app.goo.gl/9djswJZAE4rszYPH7
Hopefully others can suggest somewhere a bit more "pubby". I've never really been a "pub for food" kinda guy.