In general the research has indicated that it doesn't really play much of a role but I think at the extremes it does. Travis Green last season was the coach with the most fastidious attention to deployment in the NHL, and possibly the most we have ever seen. His players litter the leaderboard at both ends, with Gaunce, Dowd, Archibald, and Sutter ranking [1], [2], [3] and [4] at getting the lowest % of O-Zone starts in the entire league. If you lower the TOI threshold then Chaput is in there too. At the other end, you have the Sedin twins leading the NHL in most O-Zone starts by a pretty significant margin as well. I don't think there is a coach in the league who was as steadfast in his dogma of "in case of situation X, play player Y" and I think it's going to be difficult for anyone to score when they are pigeonholed that dramatically. There is just nothing else to really even compare it to.
Nobody else in the NHL was under 20% and the ones who were even close are Carter Rowney (1 ES goalsin 44 games,) Jay Beagle (6 in 79,) Casey Cizikas (6 in 64,) Steven Santini (2 in 36,) Cal Clutterbuck (7 in 76,) you get the idea. These are guys playing 12-14 minutes per game keep in mind as well, not 18 like Sutter.
If you look at the veterans on that list, guys like Clutterbuck and Beagle were scoring more in previous seasons when they were getting 40 instead of 20. I think that Sutter is something of an outlier here, and I think it's a combination of him having a legitimately great year as well as a few flukey things like empty netters and OT usage.
I think the impact is relatively minor, in general and don't think you are going to see much difference when the player is between 30-60 (let's say,) but I think at the absolute extreme ends it's basically impossible to score goals when you are playing 12 minutes AND getting 15% O-Zone starts. In that regard 4 goals in 37 games is actually damn good.
It will be interesting to see if Green changes his approach next season when he no longer has the Sedins to coddle. Does he similarly give Boeser the 75% O-Zone starts he was giving the twins or does he go to a more even spread? Maybe
@Bad Goalie has some thoughts on whether Green was this extreme in his deployments in Utica or if it was more of a matter of him being terrified of using the Sedins anywhere but the offensive zone.