I can't believe the premise isn't being examined a little more thoroughly.
Over the years, and during the entire First Dynasty, especially year 1, the Patriots were the team that was going to beat you whatever way you left open. Brady could be dink-and-dunk man or the mad bomber (once the targets were here.) The D didn't consistently totally destroy all opponents, they just always beat you.
But in that whole time, everybody talked about Brady as pretty much being the franchise. Other than that we "didn't have an identity..." and I am okay with that.
If we have no identity except that the team always wins, that is fine. I think actually that's closer to what you want. Call yourself "Blitzburgh" and people figure out how to beat your blitzes, and so on. Be dependent on one guy, and you lose if they can beat that one guy. The fact is that the Pats were not that thoroughly dependent on Brady (see Cassel, M., 2008), but of course any Brady-led team has him as the identity or a big part of it.
While I do not believe that every HC longs to have a canned narrative about the team, I do believe that every HC want to know it does some things extremely well -- the more of them the better. Because you can't have everything, the poor running team with a great QB accepts the "explosive passing attack" tag -- and the reverse. Or if the O is hitting on all 8, then it's an "explosive offense" without mention of the D. Or if you always gut it out somehow (most years, the actual style of the Pats, although Brady was always toward the top in QB metrics), that became your identity.
So for calling on the team to have a better defined identity, sir, I award you no points.
You're expressing, another way, that we're not, to date, really good at anything. We don't really scare anybody outside of Cleveland and Detroit, which is sort of like not being able to beat the local cub scouts but beating the hell out of the Brownies. Wow, Brownies... not even on purpose! But you get it... I think the problem is we're not good yet, not that nobody can say, "Oh those New England Patriots. They don't always ______, but they sure can _____"
Solution: Not "have an identity," more like "Score points, and get the other guys not to." Be good at football.