Any player on the descending arc phase of their career is going to do whatever they can and whatever it takes to stay in the game at the highest level they can, until they retire.
Henry would sign to compete for playing time. Forget “backup TE” as a role definition. He’ll split playing time, the way Hooper did. How the split goes depends on how he’s playing, how the other TEs are playing, and what particular characteristics they bring and are needed.
Remember, Hill will be back in the mix next season. I was hopeful he would grow into a bigger role this season, but that was derailed by his injury. Hopefully he’ll be able to come back from it and contribute.
EDIT to add, doesn’t Britt play LB? Or is there someone I missed?