Plus VrabelOff the top of my head, some deserving players from this era (in no particular order):
a random QB
Solder (probably a long shot)
Solder was here 8 years, started for 7, I think he is borderline
Gilmore, if he plays out his contract, will have been here for 5. I think he would need another contract.
Vinatieri is not in yet, since he's not yet retired.
And Gost may have a claim as well.
This list easily takes us into the 2030s.
Edit: and then you have guys like James White, David Andrews, Joe Thuney... if they play in NE a while longer they could all get in.