As you point out, it has nothing to do with being cheap, they spend to the cap, they just aren't kicking the can down the road like other teams who push debt into the future with future voidable years and so forth. Not only do the Patriots not do that, they even took a year (the Cam year) to balance their books.
The Patriots are like the financially conservative married professor who pays his credit card off every month, lives in a modest 1900 sq ft house that is now paid off (yea!), and whose only debt is a brand new car for the wife (with a 4 year loan). Spending to the "cap" on the wife via new car and plenty of travel, but without going into a bunch of debt or pushing a bunch of debt into the future.
Teams like New Orleans, Kansas City, and now the Browns are the married couple who when business was going good bought 3 beach condos, a swimming pool for their house, and a boat (many of the purchases with only the minimum down payment), leading to them having to refinance their house to back out the maximum amount of cash, had to sell the boat and some of the condos, and now are constantly opening new 15 month without interest credit cards and transferring their sizable, constantly increasing credit card debt to newer and newer credit cards. They are good people and have a great life, they just choose to push debt into the future to enjoy that life while they have Drew Brees or Patrick Mahomes (to bring the analogy back home to NFL Team spending philosophies).
.....By the way, I'm almost positive that my brother in law's family does not read PatsFans.com posts, fortunately for me.