The teams that come to mind for non-rebuilding cup winners, right or wrong, are the 1999 Stars, the 2007 Ducks, and the 2019 Blues, but I wanted to check that thought and look at their most recent run of finishing low. I think there are three reasons you'd associate rebuilding with an eventual cup: There are some assets that are very hard to get without a high draft pick, you don't want to let assets that aren't helping you win die on the vine, and you want an environment where players can step in and develop.
Did the Blues post lockout crash directly lead to a cup?
Did they get their hard-to-find talent by finishing low? Not O'Reilly. Not Binnington either but that's fine because goalies are delivered by storks. Obviously a yes on Pietrangelo. Tarasenko came from a lower pick, but still from a playoff miss. They had one other top pick from that era, and while he underwhelmed, he was part of a trade for Shattenkirk, who was parlayed into a draft pick and then Brayden Schenn. A bit of a stretch maybe but you'd rather have those assets than not.
Did their old assets turn into anything useful? They traded Keith Tkachuk and down the road turned that into Robert Bortuzzo. They traded Doug Weight and drafted Berglund who was in the ROR trade. Lee Stempniak is a textbook bad team scorer who raises his value on rebuilding teams and they turned him into Alex Steen.
Did they develop their players on a clean slate? They missed the chance to have Pronger mentor Johnson and Pietrangelo, but they basically kept around two decent vets (Jackman and Brewer) and other than that had miles of space for their youngsters to grow in. Johnson didn't grow so much but it wasn't because they had too many vets on the roster. I think the roster looks the same by the time Tarasenko arrives whether or not Tkachuk and Weight get traded, but I think Steen came into a roster with less clutter than Toronto had, and flourished.
Did the Ducks pre lockout seasons lead to a cup?
Did they get their hard-to-find talent by finishing low?
Lupul and Smid were top ten picks who turned into Chris Pronger. However, Chistov did nothing for them, and Bobby Ryan was still being shipped when they won. Selanne and Niedermayer were signed right off the rack, and Giguere was bought low and by this point was a fixture on the team. Getzlaf and Perry were mid to late first round picks and the Ducks were very lucky to get them for reasons below.
Did their old assets turn into anything useful? Kariya bounced, but Selanne was traded for a package that eventually turned over Rob Niedermayer which turned out to be immeasurably important. Fedorov was sold and they accidentally found a 30 minute defenseman.
Did they develop their players on a clean slate? Perry and Getzlaf reached a crucial development stage on the cleanest slate of all, the lockout, and then came into a fast league where a lot of old players were getting weeded out fast. The Ducks had little to do with that. They also signed a bunch of young-ish free wallets, but they gave them, along with Beauchemin, space to grab on.
Did the Stars late 80s struggles lead to a cup?
Did they get their hard-to-find talent by finishing low? Modano first overall and Hatcher and Matvichuk in the top 10. They were already climbing out of the hole when they stole Iginla at 11 and turned him into Nieuwendyk. Belfour and Hull were simply signed. Harvey got them 2 thirds of a checking line.
Did their old assets turn into anything useful?
Most of the good players from the 80s North Stars died on the vine (Hartsburg) or got shuffled around in meandering asset chains that did not effect the 99 cup (Ciccarelli), but you can draw a line from Brian Lawton to Sergei Zubov which is pretty cool.
Did they develop their players on a clean slate? All kinds of weirdness here, with Harvey not really panning out, guys like Zmolek and Irbe lost in the dispersal draft, etc. But the first three guys I mentioned found their role promptly enough.
I think the answer for all three is, yes, but not as directly as a textbook case like Crosby's Penguins.