I hear now from more and more Trump supporters on this point and I completely share it. It is not just Trump haters who want the President to get off Twitter, but a growing number of Trump supporters. I think and so do they that the President's popularity would go up.
Right now, the media is advancing a story that the President is out of his depth, unstable, and mentally unfit to be President. Many of his tweets amount to a verbal diarrhea of real time thinking that play into those media concerns. The President is able to advance a message on Twitter by bypassing the press, but by tweeting so much it gives the media a scattershot of moods and mindsets to cling to. The President undermines himself by tweeting so much.
I think if he took a more targeted approach to Twitter, instead of doing it like everyone else does, his tweets would have more impact on advancing his agenda and increasing his polling. If he focused on public policy instead of grievances, etc. it would force the press to focus on those things instead of allowing themselves to be distracted from the President's growing win list.
But you can't teach an old dog new tricks and I have no doubt his most rabid supporters will continue to encourage his complete engagement to his detriment. The President's enablers will be the ones who do him in, not the press.