I think Burkhead is ahead of Lewis because Burkhead played more than Lewis in week 1, played something that looked more like starters minutes in the preseason than Lewis, and because Burkhead is on the roster for the contract he's on the roster for (not that contract guarantees playing time or role, but rather that the Pats had Lewis on the roster and 1) signed Burkhead. I dont think he gets that contract or signs with NE if they weren't looking for/offering a role on offense that sort of crowds Lewis out 2) havent used Lewis and Burkhead in any way that indicates that has changed).
Yeah, its possible Burkhead becomes a ST guy (I dont think this is plan A based on his contract and usage) or someone gets hurt and Lewis ends up expanding his offensive role. I think its somewhat unlikely and sort of a luxury not worth passing up help on defense for if the stars align and the right role player is available.
I don't think you're wrong that Burkhead playing a significant role is plan A (though we didn't see that Week 1, with only 10 offensive snaps), but I think you way undersell the possibility that we need to go to plan B, C, D, etc. The Patriots basically kept their running backs healthy in 2016, but that's pretty unusual. In 2015 both Blount and Lewis wound up on IR and they turned to White and the barely-warm corpse of Steven Jackson, with contributions from guys like Bolden and Joey Iosefa at points. In 2014, Ridley ended on IR and they turned to Jonas Gray and Blount, picked up off the scrap heap. In 2013, Vereen got hurt Week 1 and Bolden played a major role. No one should be surprised if one of the Patriots' RB gets hurt.
I think you're underselling the performance uncertainty, too. Gillislee had 15 carries Thursday; he's only had more than that once in his career. His season-high is only 101 carries. Burkhead has 90 for his career, Lewis has only 151, and for that matter, White only has 80. Hell, the RB in the stable with the most career carries is Brandon Bolden. This is not a very established group, and it seems to me that it was assembled in part with the knowledge that it could play out in a variety of different ways. They may try running White more, they may try passing to Burkhead a lot, they may ride Gillislee hard, but we don't know how those players are going to respond to bigger workloads, in terms of endurance, durability, performance, or exposed weakness.
And at the end of the day, Burkhead's contract only means so much. Jonathan Cooper was making $2 MM and was also part of the return on the Jones trade; he never even dressed last year before he was cut. They paid Shea McClellin $3 MM / year and he couldn't earn starter snaps even after the Jamie Collins trade; they traded a bag of balls for Kyle Van Noy midseason and almost immediately elevated him over McClellin. They famously cut Leigh Bodden with a bunch of money left. They are paying David Harris basically what they are Burkhead; he played two snaps Thursday and I don't think anyone will be surprised if he's not on the team a month from now. Burkhead's deal says something about what the Patriots thought on him the day they signed him, but his performance - and injuries, gameplan, etc. - will dictate how that plays out going forward - and the error bars are pretty big.