Yes. And No.
Of course the Patriots wanted to win and claim homefield advantage. There has never been a time where head coach Bill Belichick said I think we should lose this game right here. Ignore any and every conspiracy theory that the Patriots wanted to avoid the Steelers in the divisional round; the Patriots were in trouble before the Jets had been eliminated to make way for Pittsburgh and Belichick has already said that he doesn't care which team he faces.
So, yes, the Patriots were trying to win. There's no other reason for trotting a hobbled quarterback in Tom Brady on the field if the team didn't want to win. Tight end Rob Gronkowski wouldn't have played 92% of the snaps if they didn't want to win. Safeties Devin McCourty (99%) and Patrick Chung (95%) wouldn't have played the whole game if the Patriots weren't trying to win.
The real question isn't whether or not the Patriots were trying to win. It's how were the Patriots trying to win, and at what cost were they willing to play?
The Patriots seemed to effectively punt the entire first half by rushing 20 times versus just 5 passing plays. It was an active clock drain.
"I think they just wanted to shorten the game with all of the injuries they have," a Dolphins coach told the Globe's Ben Volin. "And then play their regular tempo in the second half."
New England hoped that they could hurry through their first half and put the team in a position to win with a standard offense over the second half of the game. In fact, the Patriots almost went into halftime up 6-3, but a missed Stephen Gostkowski field goal led to a short field for the Dolphins to respond with 7 points and a 10-point swing. The Dolphins led 10-3 at the half.
This aligns with what the Patriots were saying after the game, per Volin:
According to several Dolphins after the game, Patriots special teamer Nate Ebner told them that New England's plan was to keep running the ball until they picked up two first downs on the ground.
"They tried to develop their running game for the playoffs," Dolphins cornerback Brice McCain said. "They tried to come pound us like that."
The idea that the Patriots offense wasn't going to try until the rushing game picked up two first downs checks out. The Patriots ran on 10 of the first 11 plays, and that passing play was a third-and-long. The team didn't pick up their second rushing first down until the field goal drive to open up the second quarter and the Patriots called 17 rushing plays against just 3 passing plays through the field goal.
The very next drive, the Patriots called 4 rushing and 4 passing plays.
Essentially, New England wanted to spend the first half of the game working on running the ball and integrating Steven Jackson into the offense. Brady was battered every time he dropped back to pass due to the makeshift offensive line, so the coaches hoped the limit the damage- and running the ball could give him extra breathing room in the second half.
Unfortunately, the Patriots couldn't pull it off in the second half. After James White scampered towards the goal line on the opening drive, and the score tied at 10-10, the Patriots couldn't function on offense. Brady threw three straight incomplete passes on the next drive, and then dialed up seven more passing plays (2 canceled by penalties) on the subsequent drive that picked up 8 total yards. They abandoned the run after the touchdown with 12 passing plays and just 1 rushing play for the remainder of the third quarter. The offense wasn't any better than in the first half.
You might disagree with how the Patriots approached this game, but they followed the plan. Fast forward through the first half, and try to win with the limited offense in the second half. Wide receiver Danny Amendola hardly played (38% of the snaps), so Brady was again throwing to Rob Gronkowski, Brandon LaFell, and Keshawn Martin.
The line was playing so poorly that the Patriots needed to use a combination of Michael Williams and a running back as extra blockers, which meant that the Dolphins could greatly simplify their defense. Miami could rush four, place four in the middle of the field to take away the crossers, and then force Brady to try and use the sidelines or deep passes over the top.
Belichick, Brady, and the rest of the team were still trying to win, but they weren't willing to risk further injury to accomplish the goal. The loss wasn't about avoiding the Steelers, or a lack of effort. The bottom line was for the team to avoid offensive injuries at all cost, and that meant running the clock for the entire first half and not rushing Danny Amendola or Julian Edelman or Sebastian Vollmer back from injuries.
They'll be back for the playoffs, the offense will change yet again, and there won't be any question about their effort.