Anything coming off the old Arthington-Menston line would need a funicular to reach the airport because of the Chevin, it is at it's steepest on that side. Also, as pointed out, already some buildings on the track bed in the Pool area.
One day, maybe in the distant future, the Menston to Otley section may be reinstated, but this will be of no use to the airport.
There is only one realistic rail option for the airport and that is a branch coming off the Harrogate line somewhere between Horsforth and Bramhope tunnel. This might have stood a better chance had the Leeds-Harrogate line been chosen to trial the new tram-trains, but this seems to have gone to a line out of Sheffield.
Personally, I would have thought the most cost effective option would be some sort of automated monorail shuttle, similar to that used between the terminals at Gatwick, to link the airport and Horsforth station.