When a player restructures or renegotiates his contract, he’s agreeing to take less for the good of the team, right?

No, in the vast majority of cases, a restructure does not mean that the player is agreeing to take less. Only in the case of a declining, overpaid veteran is a pay cut part of a restructure.

So, in the vast majority of restructures, the money doesn’t change, it is only “restructured” from a bookkeeping perspective, so that Salary Cap space is created.