Thin indeed, but that's because they have far too many services.
I would bring back BBC3, but make it a kids channel during the day, with teen stuff in the evening and BBC4 would be a combined news/documentary/politics channel, thus axing the news channel in its current form. I would cut out a lot of their radio stations which could be covered by someone else.
Problem is if they stopped the realty stuff, celebrity chefs, bargain hunt/makeover type shows, there wouldn't be much left at the moment.
If I were a BBC boss today, I would look back in the archives and remake a load of stuff and give it the HD/4k treatment. I can think of lots of really great kids shows from when I was a kid that could be remade. How about a new version of Children of the Stones, as but one example.