It would, but it would be unwise. A commuter/metro service has quite different needs from a long-distance service (this is one of the lessons of Thameslink, which does work that way, with trains running from Brighton to Peterborough through central London, and attendant compromises). The Crossrail plan is to divert the commuter services through the tunnel and free up more space in the terminals (Liverpool Street and Paddington) for the fast long-distance trains.