I think it's highly unlikely that anyone gives Edelman a big contract after next season. 31 yo slot receivers don't generate that kind of market. The obvious and most common comparison is Welker and while he got more in Denver, it was t way above and beyond what NE was offering and it was more a game of musical chairs than outbidding. He tried to use his leverage and the Pat's called his bluff because there was an obvious replacement for him in Amendola. While he is great I think his value is strongest to the Pats and their system and while it may be wishful thinking, I think the difference in offers he would get would be mitigated enough that he'd take a little less to stay on the team.
That's 100% conjecture, so it doesn't mean a whole hell of a lot, but end of day I don't see a team throwing something huge at him were he to hit FA, or at least enough that it means he leaves for purely financial reasons. I wouldn't be shocked if they used some of their cap space this year to extend him a year or two out.
Let's not quibble over what "big" is - suffice to say that I think Edelman will get offered more money by some other team when his contract expires. Said offer will be "bigger" than what the Patriots are willing to pay a 31-year old slot receiver. You are correct - Edelman's decision will probably be about more than just money, though money is always a big part of any calculus.
If there's a parallel to the Welker situation, I think it directly relates to Cooks - instead of trying to replace Welker/Edelman with a player from outside the organization at the same time as the old guy departed, they've brought in Cooks a year early. They will know, for sure, how Cooks fits into the offense/plan, and they will not have to compete with other clubs to acquire him (two distinct problems from the Welker -> Amendola transition).
Edelman may see a short-term, lesser-money deal to stay in NE as ideal; however, it may not make sense for the Patriots to want him back. Hogan is making not-insignificant money. Mitchell may be worthy of an extension himself in 2019, which would cut into any multi-year offer to Edelman. And Cooks's extension is going to be very expensive (by Patriots WR standards). Take Edelman's 2017 pay, add it to Amendola's 2017 restructure, and add Cooks's remaining entry level deal in 2017 ... and they are still short of what Cooks could (should?) command in 2018 and beyond. Add to that the fact that Gronk's contract escalates next season (as does Allen's) and the $ starts to add up.
Will Edelman take an Amendola-level pay cut to stay? The fan in me says "PLEASE" - the rest of me thinks "that's not happening".
Finally, it makes no sense to dip into this season's cap to extend a WR who will be 31 when the extension kicks in. No matter how much we love him. Edelman will not be paid for past performance and every scrap of $ lying around when the dust settles in 2017 should be applied to Cooks's signing bonus/extension.