Reason to stay in old city is because you are a few minutes walk from everything. Convenience comes at a trade-off of very slow internet and that everything is worn. Expect a issue with something if you stay anywhere in old city.
Remi's apartment is nice and basic which makes it easy to clean and less to go wrong.
Reasonably quite, close enough that each morning I could open the windows and drink coffee listening to the bagpipes bring played on Royal Mile.
Within 2 minutes walk you have several excellent restaurants, Starbucks, and many clothing and whisky stores. Within 15 minutes walk is most of the attractions you want to see. You can use bus and tram to move quickly around the city.
Edinburgh is a good base to explore Scotland by train even the Isle of Skye is only a train and ferry away. For kids it has wonderful museums with everything from Dolly the Sheep to a Dinosaur, and camera obscura.
For those shopping there is always something nearby that meets your budget, even grocery stores from Sainsbury's to Lidl tend to be close.