IMO, "lie" is a bit of a reach in terms of a word choice. It's extremely difficult to evaluate salary cap data and interpret the impact from those numbers. Unfortunately, salary cap math is not the same as real math.
For starters, Brady's departure from NE this year left the team with a $13.5 million cap charge in dead money. Put another way, DEAD CAP ALLOCATED FOR BRADY would be the 19th highest paid QB for the 2020 season if it were an actual person.
Looking over the last three NE SB winning teams, I just posted that Brady's cap hit ranked 12th, 18th, and 12th. However, if they chose to apply $4.5 million in each of those 3 seasons (to account for the $13.5M), Brady would have ranked 2nd, 13th, and 2nd in terms of salary cap hits for those seasons. If that had been the case, would people still have been saying how little Brady was playing for? Remember, Brady would have been 37, 39, and 41. To be the second highest paid player at those ages may have been considered a lot of money. He may well have been worth it, but a lot of people would have said it's crazy to pay a 41 year old QB that kind of money.
Again, it's really hard to evaluate any of this. A lot depends on when players renew or extend contracts. It's not just as simple as looking up where a player ranked in salary cap charges in a given year, as players signing new deals typically get paid better and players fulfilling their old deals are usually on the lower side of pay by the end of the contract.
Even if we say Brady took a little less and that benefited the team in some manor and helped them win more, in the long run that would have served to HELP Brady. For example, if he reached the point and said to NE I need to be the highest paid QB or close to it, NE may not have won 3 more SB's and Brady may not have been considered the GOAT. Personally, I think there would be plenty of people who would have said Brady was a good QB, but if he never won after 2004 I doubt he would have the same buzz surrounding him. I would feel confident that by giving back a few million dollars a year to NE, that ultimately allowed Brady to make 10x that amount in endorsements the past few years and allowed him to create the TB12 brand.
Certainly all of the salary cap stuff is debatable, and there can be multiple ways to look at the information and potentially conclude different things.