I live near Ealing and it's been a tremendous improvement to my commute.
You're correct, fast intercity services and the stopping crossrail services likely cannot coexist.
But technically it is possible, assuming the requisite signalling standards are met by the rolling stock intended to be used. It's possible that only the Class 345 and Thameslink's Class 700's can meet this requirement currently.
From the West end of the line in Reading nobody takes the Elizabeth Line to go into London, for instance, it's much too slow compared to the direct fast GWR trains.
This also means that the line probably cannot be shared with fast trains for direct routes from East to West, I think.