The last New York team to consistenly write stories every night would probably be the 2012 Yankees who clinched the division and won the first round hitting walkoff after walkoff.
They had a crazy comeback in the 9th inning of game 1 of ALCS and then needed a sacrifice to win that game in extra innings. They didn't get it and went onto lose. That was the night the magic ran out for that team and they got swept.
Even the 2014 Rangers, at their peak, were just clowning teams. It wasn't exactly a narrative. Sure, there's the thing with MSL's mom which absolutely created a magic moment in game 6, but if Fleury doesn't post a .972 save percentage the first 4 games, that series is over the night she died.
If anything, the Kings got those crazy games to beat us.