You're thinking of it the same way as people are thinking of Crossrail as serving Reading. It wouldn't be of any relevance to those travelling to/from Warrington, it would be a local service improving the terrible local service to many of the CLC stations.
It still might not be justifiable (I think 25kV electrification and very-high-acceleration, all-axles-powered lightweight EMUs for the Liverpool-Manchester via CLC stopping service is a better idea), but don't consider the outer terminus of a stopping train (or tram) as its primary traffic source/sink - it almost never is.
If the Airport to Warrington tram-train service used the Cadishead viaduct, the only local CLC line stations it would serve would be Glazebrook, Birchwood and Padgate. Birchwood already has a 3tph service to Warrington/Liverpool and Manchester, including a semi-fast. If the line were to loop north through Carrington to cross the Irlam viaduct, it would serve Irlam as well, but still not Flixton, Chassen Road, Urmston, Humphrey Park or Trafford Park.
I am sure that all these stations have greater repressed demand for more frequent services to central Manchester (currently only 1tp2h for some of them) rather than direct services to Wythenshawe/Manchester Airport. But tram-trains into Manchester from the CLC would have to cross to Metrolink metals at Cornbrook, further increasing frequency on the most congested part of the Metrolink network to 50tph or more (and dragging the thread back on topic

).
So, as you suggest, 25kV electrification and fast-accelerating EMUs, staying on NR metals to Piccadilly and beyond, might be a better way to improve the CLC local service.