I avoid Classic FM, ,or any commercial station, because I can't be bothered with the ads, though I know it is the only way it can fund itself. Also because the listings in Radio Times show it rather too unadventurous, sticking to the tried and found safe. It's all good music but the format and play-lists on commercial stations have to please their sponsors.
That's a bit harsh on Radio Three though. It does play complete works, and many of them; but it also plays a huge amount of music that is new, even avant-garde, sometimes commissioned; or very early, or by the less-familiar composers - music no-one else will broadcast because they want only the familiar.
It's also actively helping support working composers and musicians, which I doubt Classic FM can do.
I wonder if the RAI is doing so - from what you say it might be in its remit.