Regarding this that you said, "maybe a pear of leather boots".......
If you mean leather boots such as dingo boots or cowboy/western wear type boots with a smooth, slick sole on the bottom, then no, don't even bother taking them.
Wet pavement during a light rain will make these slick on the bottom. Edinburgh has some sidewalks that are smooth stone, plus cobblestones in other places.
These boots would be the worst thing possible to wear on cobblestones.
Edinburgh Castle has lots of stone stairs, making it dangerous to wear a boot with a smooth bottom. If you slip and fall on these stairs, you could fall all the way to the bottom of the stone staircase.
Likewise, many steep staircases in Edinburgh take you from one street down to another.
Ditch the leather boots, unless they have a tread on the bottom similar to hiking boots. Take running shoes, hiking boots, tennis shoes--anything would be better than a smooth bottom boot.