The idea of bringing in a business intelligence consultant is a good one, particularly if the person is a complete nerd and knows nothing about the NFL but has a good head for numbers and can grasp the salary cap. The Patriots do a good job of budgeting for depth within position groups, but underestimate the cost of personnel turnover on the roster, in the front office, and on the coaching staff.
In baseball, (which ought to have a hard cap like the NFL) - $32M/year for a DH?), they have a great statistic in Wins Above Replacement or WAR. It is not by any means perfect, but it does force GMs to look carefully at what happens if they dump a productive player or one with an unrealized upside. WAR stats incorporate several factors including defense and league average performance for a player in the same role.
Belichick is a WAR guy in the way he looks for players who can do lots of different things. His WAR approach is especially good with LBs and DBs who fill lots of roles on defense and special teams.
But he has blind spots - most notably WR and RB, in my opinion. For a while, there, he made some great decisions about guys like Moss, Welker, Amendola, James White, and my favorite Swiss Army Knife, Cordarelle Patterson, who is still productive with the Falcons. We've all been on threads commenting on spending money at certain positions to help an aging Tom Brady, or a developing QB like Mac Jones. I've never seen a stat like WAR for player like guy like DeAndre Hopkins, Jacobi Meyers, or Robert Woods as compared to the guys on the on roster. Maybe that kind of stat exists, but I don't see it the way we look at baseball players.