As far as the NFL is concerned, they paid him (or WILL pay him). That's why the money counts against the salary cap, and that's why they couldn't sign anyone else with the 0 room they had left.
And let's pretend that in the 1% chance they get the salary cap money back next year - how does that help winning this year (in what could be Brady's last year as a Patriot)? It doesn't. The Patriots lost the money for this year, at a bare minimum. The only opportunity they have left to improve their team this year is by signing AB to a vet minimum and agreeing to drop the signing bonus grievance (i.e. agreeing to pay him what they already owe him as reported on the salary cap).