I don't know about other Saints fans, but there was a time (back in the 70s & 80s) when my dream each season was just to see the Saints reach the playoffs. But I have moved on from that simple wish ever since the 2000 season when the Saints defeated the Rams for their first playoff victory. The bar got higher when we made it all the way to the NFC Championship game in 2006. But after having reached the pinnacle in 2009, I no longer think of 'backing into' the playoffs as a satisfying goal anymore.
So once I lose confidence that the Saints are capable of going the distance in the postseason, then missing the playoffs no longer bothers me. While I realize that anything can happen once you make it into the NFL's second season, I can't help but to view a Saints team who has struggled all season long as not being worthy of being in the playoffs.
Do I still want the Saints to make it into the playoffs even after a mediocre season? Of course I do. But I see such a season as being the proverbial 'handwriting on the wall'. I don't get nearly as excited and I 'brace myself' (translated: fully expect) to experience the inevitable early exit from the ranks of the real contenders. For me, it's just hard to get my expectations high for the playoffs when the Saints have proven throughout the season that they aren't ready for postseason glory.
Having seen my team bring home a Lombardi trophy once before has taken away a lot of the pressure & stress of seeing the Saints get back into the postseason. That's why it no longer bothers me when the Saints come close, but (mercifully) fail to make the playoffs. I want our team to be a true contender. So as you mentioned, having witnessed the disappointing losses that were a big part of the 2023 season just made our elimination from the playoffs at the end a lot easier to accept.