Looking back at results, I'm not sure how much I agree with that anymore. Granted, it's admittedly complete hindsight on my part so I fully acknowledge that. Two - three years ago, I probably would have said (and maybe even did say) similar things as you wrote above....the team is heading in the right direction. I recall there were others (Wally-
@Ole Gil being one of the main ones) who disagreed with that assessment back then and thought Peters was a terrible coach back then.
Kirk Muller, who by most accounts wasn't a good coach, coached the team to 36 wins and 83 points in the season he was fired. That team lacked talent as much as any Peter's team (maybe even more so) and had to use Justin Peters in net for over 20 games. I recall the team employing a "pack it in and keep shots from getting through" approach that season when Peters was in net. Khudobin had a good season, but had injury issues as did Cam Ward.
In season 1, Bill Peters takes over and promptly leads the team to the 2nd worse record in Carolina Hurricanes history with 31 wins, 71 points. The team went from scoring 201 goals the season prior, to scoring 188. The team allowed the same number of goals.
In season 2, the team had 35 wins, 86 points, so 1 less win and 3 more points than the season that got Muller fired. On top of that, they stunk it up early in the season and by December 1st, they were the 2nd worst team in the NHL by that time with an 8-12-4 record. They played better later on in the season, but at a time when it really didn't matter any more.
In season 3, the team had 36 wins and 87 points. So the same number of wins as the team under Muller, and 4 points higher. Again, they struggled early in the year and were 14th in the East on Dec 1st.
Peters no doubt instilled a more much more structured playing style that emphasized minimizing shots, moving the puck quickly in transition, and taking a lot of shots. The team got faster, played faster and made it tougher for opposing teams to get sustained offense, but the Canes didn't generate a lot of offense (scoring, not shots from everywhere) either. They were analytics darlings, but even though they gave up less shots, it seemed they gave up more high quality scoring chances.
In the end though, the team didn't improve much at all on the record that got Muller fired, even though the talent was similar (if not better in later years) to the talent Muller had. Don't get me wrong, all of those teams (even the ones under Muller) lacked talent badly. I don't think any coach could have gotten much out of that group of players, but looking back at the actual results, I don't know that I can conclude Peters did a great job either.
EDIT: that doesn't change or negate your point though. At the time, Peters was generally viewed as a good coach that was moving the team in the right direction, so him getting an extension wasn't seen as unusual.