I think it would be a much bigger factor if the Celtics prove they are close to being competitive with the Cavs. If they get swept, I doubt beating the Wizards in a close 7-game series will mean much at all in Hayward's thinking. The league is very top-heavy and it's really difficult to jump from the 2nd tier to the 1st. Just because we made the ECF doesn't mean we're necessarily at that tier yet.
The Celtics will have to make the case that despite Utah having some talent, this Celtics team is closer to actually joining that top tier. I would make the case that in the next few years the Warriors with Durant is still a tougher mountain to climb than the Cavs with an older LeBron.
Then I would point out that while Utah has done a great job with finding talent (especially Gobert), they still don't really have that secondary offensive star that would make his life a lot easier, nor do they have the cap space or assets to acquire one. They would have to do some kind of overpay like say, Hood, Exum, and/or Favors, plus multiple future first rounders to even start a conversation. It would gut their team of bench depth and assets for the forseeable future.
Also, as good as Gobert is, teams are able to neutralize him by going small and forcing him to abandon the paint. I would expect the Celtics to point this out in a delicate manner.