The fancy stats may not say they are any better than with Hak but I think as big a factor of having Hart in net is actually having someone on the bench that has a pulse and has breathed some life into the team! The attitude of the dressing room seems much more alive than when Hak was here.
I think that's more a reflection of Hart's impact than anything else.
The Flyers played hard under Hakstol, but it's deflating to push and get back into a game then watch your goalie give up a "softie," and when that happens again and again, it kills team morale. The team perked up when Elliott was on a roll, then when he went down and the parade of scrub goalies began, they went flat as a pancake. The only personnel change was moving Sanheim with Provorov instead of Ghost. Lindblom was in Gordon's dog house for the same reason as under Hakstol, he had a stretch where he wasn't playing well.
With Hart, the players can be looser, because every mistake will not end up at the back of the net.
In fact, sometimes they may be too loose (Gordon has complained about, sloppy defense), they giving up a lot more scoring chances and high danger scoring chances than they did under Hakstol (from top 5 to middle of the pack).
A lot of the complaints about Hakstol are really wishful thinking, if everything was his fault, a "magic" coach will solve all the problems of this team. People ignore that the Blackhawks had 76 points under Q last year with players like Kane and Toews and Keith on the roster and goalies putting up a .911 S%. The Flyers, with horrible goaltending, were on a 74 point pace under Hakstol.
Hart has solved the biggest problem (at least for now), the maturation of young players on the roster and the addition of a few more will solve other problems, and a couple judicious moves by Fletcher (more than just dumping scrubs) will finish the job.
A better coach will help, but as the Oilers have shown, it's not a panacea.