Like others have said, I don't believe that Belichick has ever been shy from sitting someone if there's better talent waiting to see the field, despite what they're getting paid.
I am assuming that the coaches (mainly singular as in "coach") feels that Amendola is still the best option to do what is asked of him, partly due to the lack of better options behind him, and partly due to the fact that they likely still feel that he can produce.
The key is exactly what @mgteich has stated now several times in that our WR3-4's have never produced much. With Edelman and LaFell (not to mention Gronkowski and even Vereen), a guy like Danny Amendola isn't going to receive many reps, and one would have to assume that the coaching staff knew that way back in the offseason; yet they still kept him for reasons that we do not currently know.
Look at guys who have played the WR3 role recently:
WR3 production
--2013 Dobson 37 catches (which is high due to the loss of both TEs + poor options/injuries)
--2012 Edelman 21 catches
--2011 Chad Johnson 15 catches
--2010 Brandon Tate 24 catches
--2009 Prior to the drafting of the 2 TEs it jumps back up to 37 with Edelman, which is where it was last year as well, but most years it isn't going to be as high as the mid-late 30s.
This is a pretty good perspective. Amendola has shown he can produce when called upon in a bigger role at times last season. Edelman's opportunities (and production) were limited when Wes was here. There are only so many passes to go around.
So when people point to the number of catches, it's a foolish metric. It's like watching a back-up NBA player get to start, and collect way more rebounds than usual. Well yeah, you will accumulate those things as you get more opportunities.
The reality is most teams don't get a ton of production out of that #3/#4 WR spot. There are also fewer opportunities when you mix in the TEs. So let's include those guys. I'm not including RBs, who are often in the top of the mix but usually as check-downs or last options (though Vereen might be the exception to that rule). That would make Amendola 5th on our team in terms of primary pass catchers. Let's look at other teams in the top 10 in passing attempts per game and see where their 5th guy is at. Below, a breakdown in catches, yards, TDs.
1. Colts: Donte Moncrief. 17, 218, 1
2. Saints: Josh Hill. 9, 122, 3
3. Eagles: Brent Celek. 19, 206, 0
4. Broncos: Jacob Tamme. 6, 57, 2
5. Raiders: Denarius Moore. 12, 115, 0
6. Steelers: Lance Moore. 10, 141, 2
7. Patriots: Danny Amendola. 9, 82, 1
8. Lions: Theo Reddick. 15, 154, 3
9. Falcons: Levine Toilolo. 15, 149, 1
10. Bears: Santonio Holmes. 8, 67, 0
Also including two teams who are near the bottom in attempts per game, but in the top 10 for most passing yards per game (the rest of the top 10 is similar to the list above).
19. Redskins: Jordan Reed. 22, 207, 0
26. Packers: Richard Rodgers. 9, 117, 0
The Colts, Eagles, and Redskins jump out on this list for production, although 115 of Celek's 206 yards came in his last game. Reed is a bit of an outlier as he missed 4 games due to injury and should rise up the charts if he stays healthy.
Amendola's production hasn't been incredible, but it's in line with most of the top passing offenses in the league. He's obviously an expensive option (although surprisingly not the most expensive player on this list), but he also provides quality depth. If Edelman were hurt, he could capably fill that spot. You couldn't say the same for a lot of guys on this list if one of the top WRs went down.
At the end of the day, there's nothing more left to say or do. It is what it is. He's paid more than he's worth right now, but he could be worth every penny if something bad were to happen. He hasn't produced like we (and he) wanted, but part of that is the emergence of other players. His contract didn't prevent us from doing anything so there were no opportunity costs, he's one of the best 53 players and deserves a roster spot, he's provided decent production considering opportunities (not even going into the bad luck on penalties on his catches).
And quite frankly, he's a New England Patriot. Whether you love him or hate, it's time to shut the **** up and cheer for him. All this ******** back and forth posting isn't going to change BB's mind. So be a fan, root for your team, root for him to do well and make plays to win games. It's too late to do much else.
Maybe cheering for him will make a difference. Maybe it won't. But it sure has more of a shot of helping than 30+ pages of anger and frustration and constant venting on the poor guy.