there are other routes from one side to the other. North and south of the city centre…
Not really sure what you’re suggesting in that case – the buses that do go through the city centre mostly go down Princes St because that’s the natural route, and that buses that don’t go through the city centre already don’t go down Princes St because they don’t go through the city centre!
And if you routed buses that currently go down Princes St so they ran “north and south of the city centre”, then they wouldn’t go through the city centre any more, which is no good for people who want to use them to go to the city centre.
Edit: by which I mean, if they go through the centre, Princes St is going to be a natural or even unavoidable route for them. And if they don’t go need to run through the centre, they don’t need to go down Princes St, but Edinburgh’s a pretty centralised city and I suspect most of the people on existing bus routes through the centre do actually begin/end their journey in the centre. But I might be wrong.