There's a couple things going on here one, our PP system actually discourages shots by the Dman from the point, opting instead to get shots from the forwards far more, Chabot for example sees his shot rate drop about 40% when on the PP, down to about where Brannstrom's shot rate at 5v5 is, so while you've grouped PP and ES to try and make it seem like the comparison should favour Brannstrom, including PP actually hurts Dmen shot rates in our system.
At ES, Brannstrom is second on the team in Shots/60, behind only Chabot, Brown is actually 3rd on the team.
League wide, Brannstrom is in the top 90 in sh/60 at ES (300+ mins played), Josh Brown comes in a 133 out of 212, which is actually better than I'd have guessed he'd be, he's definitely shooting a lot more this year than last year (Brown ranked 209 out of 209 in sh/60 with 300+ mins played last year)
If I had to guess, I would think Brannstrom is more likely to maintain his shot rate going forward than Brown, he's been more consistent in that regard over his career. Brown's elevated shot this year seems anomalous.
One last thing regarding the bolded claim,
Brown's shot through rate (shots on net divided by shot attempts) is actually worse than Brannstrom's at ES,
Brannstrom has gotten 54.5% of his shot attempts on net while Brown has gotten 49.1%