And yet, every single piece of evidence we have says that's precisely what happened.
E.g.,
an article from the time based on an interview with Welker's agent:
All accounts I've ever read say the same. Even if we assume that every sportswriter was identically mistaken on that front, we do know pretty certainly that Welker had visited the Patriots and the Patriots were preparing an offer sheet. But they didn't want to wait out the Dolphins right-to-match period, and/or they were nervous about the then-murky poison pill rules. So instead they offered an extra draft pick to pry him away from Miami quickly and cleanly. HOW?
Welker never received an offer sheet from the Patriots, and it was very much NOT in his interests to sign his RFA tender. Even if he ultimately signed the tender to allow the trade to go through, it could only have been after the trade negotiation was completed.
I'm honestly baffled by the whole thing.