Note: Cross-post, but directly appropriate to question of O.P.
Yes. I think this whole thing is a similar public relations agent move from BB as just shown in the HT situation.
Firsty, the Cooks deal is a separate entity. No hidden promises, 2 for 1's, winks-winks or anything like that. Not BB's style, and too many variables or newly interested 3rd parties butting in that could go wrong for either team.
Butler's agent did not believe the offer the pats gave his RFA client was fair/market value. BB said, well then go work out a deal with an interested team and see. Obviously any such deal would have to factor in the loss of the team's first rounder. Just as BB had factored in the extra cost with his offer. Perhaps which, the agent didn't quite understand or value the pick highly enough. NO had just shown interest, so logically that's where they started, and off they went to make a deal.
And so, they did. The agent went to NO and said what type of contract would you give my client? They hammered a deal out. Probably right in line as everyone has said, in the 7M-8M AAV 3-4 yrs range factoring the loss of the NO first, the 11th.
Why wasn't a deal done? They know the offer is fair/market, just not worth signing. More importantly for NE, and as planned, they now also know BB wasn't screwing them with his offer either. It is what it is. Take a now-known-as-fair fair RFA extension deal or play on the tender, assume 1 year of risk, and get the big UFA deal next season. From the looks of it they're going to roll the dice and cash in as a UFA. Or, maybe try to find a team with a later 1st rounder with a more tempting offer, but they know that's not very likely to really move the compensation needle satisfactorily.
In any case, for the UFA plan to work and succeed you need to be a good soldier. If you're planning on shooting your way out of town as a RFA forcing a trade, sitting out or pouting, you don't go sign some autographs and say you want to be a Patriot for life. So I think what we're now seeing is a more realistic Butler, less worried about, or not believing in a fair market RFA offer, playing nice, here for another year and about to play some serious contract year ball paving the way for his big UFA deal after another high profile Super Bowl run!