Hanna has been here 4 years, played in 78 games, albeit not very much. Swaim has been on the team 3 years and played in 28 games.
Someone doesn't have to be a starter or play 40 snaps a game to "see what he's got". In addition to regular season games, there are pre-season games and camps. I suspect the Cowboys DO know what they can do, which is why they are not getting more snaps. And it's not just catches, it's blocking too.
If they were keeping Jason out of sentimental reasons, which I highly doubt, he'd still be on the team but would not be the starter. And they don't run a tight end-based offense, so his "lack of production" is likely due to the offensive scheme, rather than his loss of speed (which he never had a lot of anyway).
Jason is probably doing as much as they want the tight end to do, as well as needed...