I see there being little motivation for NE to pay competitively in the backup market. Given the money that they would likely be offered by BB/NE in light of the fact that it would be a short-term arrangement under which they would almost certainly be cut as soon as Brady returns to the active roster, I also see very little motivation for any QB good enough to stay on a team for the whole season to sign a deal to "compete" in NE this year. So, yes. Taking a pro-rated salary for 4 weeks plus a minimal signing bonus in a league where solid vet backups make millions?
The only counterpoint to that would be assuming they would want to trade JG in-season before the deadline and keep this grizzled veteran, but even then they'd also have to be prepared to take the cap increase comparing JG's cap hit to the vet minimums.