Being slow, they could likely run back via Richmond or via Wimbledon - or both. They could all be loopers.
I really don't get the airport to airport demand, it's super niche and a weird fetish on here. Virgin Atlantic people had the honour of being based on Crawley (having since left LGW during the pandy, annoying for staff!) but most flying from LHR. But that is so edge, and they put on buses.
Demand-wise, airport rail should be modelled around staff. Traffic and decarbonization - passengers, i.e. less cabs and kiss n fly. But connecting passengers, non-existent. I would think domestic visitors to 'airport hotel conferences' would rank higher!
As for the Southern Rail Link, it's solid but I understood that heading west would be a non-starter due to the various level crossings at Egham and Wokingham. If those are discarded, these plans look great. If those still matter, we're headed towards Feltham only. In that case, I'd think 2tph via Hounslow good for local journeys, as well as the main services via Richmond to Waterloo. Question - what is the minimum tph going to be to justify an investment. I would think nothing less than 4tph, but ideally 6tph.
HEx/Crossrail to Staines, I could see an additional 2tph using the link, with wires all the way to Staines (where bi-mode/third rail switch would need to occur for the SW services)