Yes I can't see HEX surviving after crossrail.
Everyone I know that takes HEX gets a taxi at paddington. Even if you're not cost sensitive you might as well beat the brutal central london traffic and almost certainly be closer to your destination.
Assuming there is good branding at LHR for crossrail so people know it exists...
It is likely to need to change its business model, but I wouldn't be surprised if it kept going for sometime after Crossrail opened.
As I've said before, the paths are useful for trains approaching Heathrow from the West, in that the western services effectively use the same paths as the eastern ones. As such the services aren't a waste of paths (even if it you looked at them in isolation and ignored the future link it could like that now).