To limit the cap hit on the last contract Brady signed, NE added a dummy year to his contract (the 2020 season) even though the contract itself guaranteed the contract would end after the 2019 season. So NE had until the first day of the 2020 season (March 17th I believe) to negotiate an extension with Brady, if not he became an UFA. Since they didn't get a new deal done, they immediately had to take a $13.5 million cap hit charge for the previously unaccounted for cap balance. Had they negotiated a new deal, NE would have been able to continue to roll over some of that cap hit and reallocate it as part of the new deal.
As for your question, yes he would have had a higher cap number if he resigned with the team, but they could have gotten more creative cap accounting wise. If they had come to a one year extension, then they could have split that leftover cap hit over two seasons. I believe if they agreed on a multi-year deal, then they could have played around with it even more. Teams do stuff like that all the time, essentially kicking the cap hit down the road. But at some point teams still have to account for it, and this was the year NE had to eat the cap hit.
On the flip side for NE, they get out of their dead cap situation and get off of some high priced contracts after this season and are set to have $106 million in cap space heading into 2021 (less whatever they use the rest of way this year). Between this year and next year, they are also projected to have 25 draft picks. They also have most of their draft class from last year that they essentially redshirted and have a ton of players that went on IR last year coming back. Between those two groups, IIRC, that's like 12-15 players. The point being, NE has a chance to completely reshape their team, get much younger, and go free agent shopping next year. So they will likely be in a down year this year, but they should be restocked and ready to roll again in 2021.