A couple of points -
a. I think you are being harsh about the Pats seeming lack of urgency in the FA market at TE. You first have to wonder just how much of a surprise Gronk's retirement was to the FO. Don't you think that it is very likely that Gronk gave them an indication of what he was thinking BEFORE his announcement. So if Cook really was high on their priority list they might have gotten him....but at what price?
b. Even if they knew Gronk was retiring, and then assuming the Pats were ready to match the Saints offer or even go a bit higher, who's to say Cook would come here. The Saints offer the same, if better prospects of a winning season. He could live in NO, or Foxboro? Play in a dome or outdoors in December, etc.
c. As far as the draft goes, I see Fant and Smith as HB's and I'm not spending any of my first 3 picks on an HB. In my ideal draft, I'd be getting Sternberger in the 3rd, and Warring in the 4th or 6th. Wesco is also an option as a blocking TE who has good hands and slow feet. Any two of those 3 would be fine.
d. As for any trade or FA. Jenkins is a non-starter. I liked him coming out, but really hasn't shown the fire that they look for. here. Shaheen was another guy I liked coming out, but I don't really have any idea how he's played now that he's in the league. Maybe he's worth a 3rd day pick.
e. It's a fact that BB's LOVES TE's and believes they are critical to the Pats offense. He's he's used several first and 2nd round picks looking to find QUALITY TE's, and is willing to swing and miss to try and get one. Graham, Watson, and Gronk were hits. We all know the misses.
f. Assuming after all is said and done and what we have left is a group of moderately productive TE's, then the Pats will simply change the narrative of their offense until they DO find the next impactful TE.