I think Welker mentioned in his A Football Life episode that he took the Broncos' offer back to the Patriots to see if they'd match but they had already moved on. Welker was as far as I recall the classical case of Belichick / Patriots setting a max value for a FA and then moving on to the next target (in this case Amendola) once it became clear that the FA wouldn't take the offer.
The issue with Welker already started back during the 2011 season when the Patriots offered him an extension of 2 years at 16 million guaranteed but he turned that down as he wanted a longer deal. In the end he played 2012 on the franchise tag for 9.5 million, before signing a 2 year deal worth 12 million with the Broncos. The Patriots stance of not giving him a 3 year extension at that (or similar to that) AAV in 2011 was probably justified considering how he fell off in 2014 (although the big elephant in the room of course is how much Manning putting him in harm's way contributed to that), although it potentially cost the team a Super Bowl appearance in 2013 (however, I don't think they beat the Seahawks without Gronk assuming in this scenario he still gets taken out by T.J. Ward).
Ultimately, however, it worked out for the Patriots as his departure enabled Edelman to step up, while Amendola - even though he never truly lived up to the deal he was signed to - was a clutch player for 3 Super Bowl bound teams.