The Patriots generally pay market value for players, they just don't do it in unrestricted free agency all that much. The idea that every Patriots player plays below market value is mythical; Brady does, but he's a unique case. The other ones currently below market value will be coming up for renewal relatively soon. These guys are workers, they care about money. There may be a few other concerns, such as moving a family, but for almost everyone those are secondary.
As for Bennett, it doesn't matter whether it was real or merely perceived because it's what he was explicitly frustrated about. The Bears also admitted to making up a rib injury so they could stick him on IR, so we shouldn't pretend his perception wasn't rational.
I never said every Patriot plays below market value. Brady isn't the only Pats veteran that signed a contract for significantly less than market value. Chris long could have made much more with another team. That said, I don't believe that their are a lot of players who take "home town discounts".
The Pats establish their own value for players, and that rarely exceeds market value. The Pats have been in situations where they paid what I would consider "top dollar" a few times. Such contracts have traditionally been given to consistent, good locker room guys that are relatively durable and play a critical role in the schemes the Pats wish to run. Logan Mankins, Vince Wilfork and Devin McCourty are recent examples. It is worth noting that the Pats didn't pickup Wilfork's option for another year and Mankins was traded.
Teams that are desperate for a player will pay "top dollar", which, IMO, is more than market value. Often times, teams that are desperate aren't attractive landing spots for a variety of reasons. Many players that sign contracts probably could have played ball a bit harder and got more money at a more desperate team. Aspects like location (NYC vs. Green Bay for example), team culture, winning, state taxes, friends on the team, projected role and stability of the coaching staff are significant factors in many players' decision making.
I don't want to get into a semantics debate that "top dollar" is market value. It is a logical distinction that can be useful in analysis, and to deny such a thing simply make's one analysis less granular.
Assumptions can be rational, and are often true, but they are still assumptions until they are proven.