Not that I would know, but I imagine that even really really smart people can make a decision which they believe is based only on fact and observation, but maybe is tinged with an overlay of emotion, or maybe they did not adequately factor in unintended consequences of the decision. .
Not giving Butler one defensive snap after watching his game plan and replacement rotation get routinely torched seems more emotionally based, than one based in correctly assessing reality.
I don't think the decision to not start Butler was emotional. Bill and Patricia (as he was most certainly involved in that decision, no matter what the radio trolls claim) decided based on the evidence in front of them that Butler wasn't going to be their best option for starting CB. I believe all the factors that *may* have went into that decision have been discussed already, so I won't rehash them, other than that it definitely appears to be truly based on football reasons, not hubris, emotion, arrogance, or Butler's contract (note: I'm lumping in any possible disciplinary issue into football reasons here, as they would be closely intertwined).
The decision to not sub Butler in when things were going badly is one that is well worth dissecting. That is where perhaps some "unconscious emotion" was in play. Let me try to explain: Belichick and Patricia are prepared to play the game without Butler, barring, of course, injury. So, at the strategic level, there is never any thought of "Let's play Bulter in the event these things start happening....". So, when the adjustments are made, they never include Butler, even if evidence was mounting that Butler may have helped. There was some deep rooted bias, a bias of which they were unaware, that simply prevented them from doing the necessary outside-the-box thinking that would have been required. This is a bias that happens to the best organizational leaders, so Belichick would not necessarily be immune to it.
Then there's the tactical situation. First drive starts with a couple of 3rd down conversions by the Eagles, but then, helped by a false start, the Eagles have to settle for a FG. OK, not a disaster yet; coaches are thinking that had the players executed on those two 3rd downs, defense would have gotten off the field. No need to break the Butler glass.
Second drive goes poorly (Philly TD in 3 plays). First adjustment is then made to have Gilmore cover Jeffrey from that point forward, IIRC.
Third drive is a 3-and-out. Butler comes in with the punt coverage team. But the coaches are perhaps thinking their adjustment worked. So give it another go without Butler.
Fourth drive is another disaster, which ends with Blount going 21 yards for a TD. OK, this is interesting. If I want to go into 2nd guessing mode, it was after this drive that an adjustment should have been made. However, I haven't rewatched the game, certainly haven't looked at film, so it's possible that adjustments were made that we just didn't see. And the game is still well in reach despite the Philly TD.
Fifth drive features a 21 yard run by Ajayi, but then ends in an INT. Yes, it was a fluke, but still, the coaching staff is still thinking that the new Rowe/Gilmore assignments are possibly working (2 stops in 3 drives).
Sixth drive is where they got bit on the wheel route to Clement. Not sure if Butler really helps that. Then, after the D did their bend-but-don't break routine for 3 plays, they get fooled by the pass to Foles. That's not really a personnel issue as it was the coaching staff just getting fooled.
Now we're at half time, and this is probably the next major opportunity to make any subs. The problem is that Rowe and Gilmore are not playing badly, and so they would be taking a bit of a risk by inserting a very emotional Butler into the CB2 position, and moving Rowe to the slot. Maybe they thought that if they could execute better on the next wheel route, they could at least contain the Eagles somewhat in the 2nd half and win the game. This is where those internal biases are possibly coming into play.
In the first drive of the 2nd half, the bottom falls out as Blount and Ajayi gash them up the middle, and Foles finishes by throwing a perfect strike to Clement, a pass that could have been ruled incomplete. By the time Philly gets the ball again, the Pats are only down by 3, and it's very late in the 3rd quarter. At that point, it's getting a bit late to take a guy that has played 0 snaps to that point and throw him out there. I won't bore anyone with rehashing the final 4 Philly drives.
From what I can tell, it does appear that tactically, the game may have gotten away from them, with the coaches ringing the same bell, thinking "We just need this unit to make one more stop on 3rd down, which we know they can do, despite the fact they haven't done it all game.".
Not defending Belichick and Patricia here; it's their job to overcome both their biases and the game situation and make the correct adjustments. Just trying to come up with a possible cause for the lack of in-game adjustments beyond "hubris and arrogance" or "idiocy", neither of which I feel apply.