I think it is station logistics that will do for the Heathrow Express.
Currently the plan, AIUI, is for four Crossrail services per hour to Heathrow, replacing Heathrow Connect and the HE continuing to run to Paddington High Level.
That means one every 15 mins, so you end up with the central core stations getting clogged up by passengers with large amount of luggage on the platforms, letting up to five trains go past while waiting for a Heathrow train.
Get rid of HE and you can have eight an hour from Heathrow to Crossrail.
In the longer term, once the western extension runs from T5 to Slough, I can see the whole suburban service from Paddington running via Heathrow, with Iver and West Drayton getting services in peak hours only.