The RS guide suggests either Old town or the Southside BnB's, but we are finding the suggested inns either booked or too expensive for our family. We're also finding a few moderate priced inns and hotels listed in New Town and West End areas that are still within 1 to 1.5 mile radius of the Royal Mile. Any reasons to not consider these areas?
None whatsoever - both areas are fine.
I have been to Edinburgh a few times and stay in New Town. Nothing wrong with it at all.
We stayed at the Ben Doran guesthouse on Mayfield Gardens, which is the main north/south road that bisects the Royal Mile about 1.5-2 miles north of the guesthouse. There was a very convenient bus stop just outside the front door that took us straight to the Royal Mile and would have continued on to the train station and New Town. It’s an 18th-century townhouse with a beautiful sweeping staircase and gorgeous rooms. The entire block is full of similar old townhouses that are now B&Bs, with lots of restaurants nearby. We were even able to park off the street in their front yard/driveway.