This is a really obscure thing to care about.
It's hard to be 100% accurate without re-watching every play from the past few seasons and putting together a play database, but we can get some measure based on Brady's splits and passes behind line of scrimmage. This is not a perfect measure; some screens may go past the LOS and fall in the 1-10 yard pass range group. If anyone has an issue with it, feel free to do the breakdown and share the data
Tom Brady's recent splits - Passes behind line of scrimmage
2011: 73.3%, 75 attempts (12% of pass attempts), 6.25 YPA, 1 TD, 1 INT, 88.1 RAT
2012: 75.0%, 96 attempts (15% of pass attempts), 6.29 YPA, 3 TDs, 1 INT, 96.9 RAT
2013: 75.3%, 93 attempts (15% of pass attempts), 5.20 YPA, 1 TD, 1 INT, 85.6 RAT
2014: 82.7%, 81 attempts (14% of pass attempts), 4.09 YPA, 0 TDs, 83.7 RAT
2015: 81.5%, 108 attempts (17% of pass attempts), 5.61 YPA, 3 TDs, 99.3 RAT
2016: 82.4%, 74 attempts (19% of pass attempts), 5.11 YPA, 3 TDs, 101.5 RAT
Looking at the past 6 years which includes 2 trips to the Super Bowl, we can see that both years we went to the Super Bowl featured the least amount of screen passes as a percentage of attempts. So I'm not sure what the significance of screen pass attempts matters in the big picture.
BUT
If you actually look at the breakdown, this season has had MORE pass attempts (as a percentage of attempts) from Brady than any of the previous 5 seasons.
That's not including Jimmy G, who had 24% of his pass attempts as screens, or Brissett at 29%.
So if anything, the exact opposite of the OP is true, we have thrown MORE screen passes this year than any of our previous 5 years.
The only thing I can think of is that Jimmy and Jacoby were throwing so many screens, that by comparison, Brady doesn't seem to be throwing as many as his backups. But he's still throwing more than he has in the past.
So I don't really understand the issue, or even care if we throw more or less screens. It doesn't have any real connection to winning or losing or Super Bowls. We might as well be debating the number of post patterns run or times we used 3 safeties on the field in the nickel, as if it will reveal our chances of winning another championship (it won't).