I really doubt the Piccadilly Line would be able to handle more passengers. I think only the Elizabeth Line would be able to be extended.
I'm always mystified by this notion that the Piccadilly Line can't be extended past Heathrow due to capacity issues. The capacity problems are eastward from Heathrow to London. Any extensions westward would serve local demand to and from Heathrow, and not be clogged up by the badly informed taking the slowest possible route all the way. My thoughts have always been on extending the Piccadilly to West Drayton, with Uxbridge as an eventual target.
As for a Staines connection, just extending the Elizabeth Line there would tap into plenty of transfers off the fairly dense SWR network. Services shouldn't go further until big infrastructure improvements happen further out.
As a regular flyer to Europe, I always appreciate the few airports there that are truly plugged into Intercity rail services. Frankfurt and Schiphol are the best I've used. Cologne/Bonn has a station on a high speed line but but I've never used it. Dusseldorf airport has the station but Intercity trains don't stop there!
Munich is a busy airport, but it's too far out with only slowish S-Bahn services connecting to Munich Hbf.
In the UK, Birmingham Airport is fairly good for trains.
Coaches to/from Heathrow are generally quite good. I often use the two Railair routes from Reading and Woking. The former is much more reliable now that the M4 is a "smart" motorway as far as the A329(M) junction. However, the Woking service is plagued by M25 congestion and any roadworks on the normal road network (especially as it serves Guildford too). The Oxford coach is normally fine. The waiting area for coaches/buses at T5 is appalling. Heathrow needs to invest in a proper comfortable waiting facility like that at Central Bus station. Even there, an extra departures board would benefit the 50% of seated passengers who currently can't see one!