Here's why the Wings 90s/00s were an impressive dynasty:
From 1991-1992 to 2008-2009, the Wings finished top 2 in their Conference in 16 of 18 seasons.
Back when the 50's Red Wings dominated, they had to win 2 playoff series to win a Cup. In other words, top 4 teams made the playoffs.
Imagine if the 90s/00s Wings only had two play 2 rounds of playoffs to win a Cup? They'd have been in 16 Conference Finals and rested going into all of them rather than coming off a brutal 7 game Semifinal like in 1996 with the Blues(for example).
How many more Cups do the Red Wings win if they play in 16 Conference Finals? As it is they only played in 8 Conference Finals during that time.
Interesting thought experiment. Applying the new playoff format to all those seasons, with my best guesstimate as to whether or not they win the Cup:
1992: Hard to say. The four playoff teams are completely different from the four actual semi-finalists. They'd have played Vancouver and then the winner of a NYR/Washington series. I would favour the Rangers in this scenario.
1993: Not likely. I really doubt Pittsburgh loses to Boston in the new Wales playoff. Detroit would play Chicago, who would have home ice advantage. I'd favour the Penguins in the new scenario.
1994: I'd lean towards no. They lost to San Jose, so you'd have to think Toronto (who beat San Jose) has a good chance of upsetting them. Nothing really changes in the East, so Rangers await the winner and would be favoured.
1995: Probably. They play St. Louis (real-life first round loser) in the semi-final, and you'd favour them heavily. Instead of New Jersey in the Final, it's Quebec or Pittsburgh. Detroit would be the clear favorite.
1996: No, they lost to Colorado anyway.
1997: They miss the playoffs, so one real life Cup just disappeared.
1998: Probably. They play Dallas, just like in real life. Instead of Washington, it would be New Jersey or Pittsburgh in the Final. Devils would probably have been a tougher test than the Capitals, but this wasn't an overly strong NJ team.
1999: Miss playoffs.
2000: It's possible. They would play St. Louis, who did get upset by the Sharks. The East wouldn't change from real life, so a very good Devils team awaits in the Final.
2001: Doubtful. They'd need to go through Colorado then New Jersey (if we assume the Devils beat the Ottawa team that Toronto swept in real life), so they'd be underdogs.
2002: Very likely: They play Toronto or Boston in the Final instead of Carolina, but should have rolled over either of them.
2003: Maybe. They'd play Dallas and presumably New Jersey, but these were slightly weaker Stars and Devils teams than a few years previous.
2004: Maybe. They would have played the Sharks, then probably Tampa Bay. The Red Wings were starting to look old by now, I'm not sure they survive two younger, fresher teams.
2006: Who knows. They'd play the Dallas team that also choked in the first round in real life, and then a Carolina/Ottawa winner. I'd favour Carolina since they pulled it out in real life.
2007: Hard to say. Semi-final sees them play a Nashville team that fizzled out pretty fast in the first round in real life. Buffalo/New Jersey would be the East Final. I'd say they end up in a very close Final against the Sabres.
2008: Probably. They'd play the same Sharks team that they beat the year before (in real life), and they'd play Pittsburgh in this fantasy final as well (assuming the Pens beat Montreal).
2009: Maybe. It would be a rematch with San Jose, but Pittsburgh misses the playoffs in the other conference. Washington or Boston would have been the East representative.
Conclusion. They should still win 1998, 2002, and 2008. They don't win in 1997, but they probably win one before that (1995 looks like the most likely candidate). 2003-2007 is the most advantageous stretch of this fantasy scenario. They probably win a Cup in that span, maybe a couple. Assume 2008 and 2009 combined still yield a Cup, like they did in real life. If I had to choose a number, I'd say they win 6 Cups. So more than in real life, but still less than Montreal won during some of their best 17-year stretches.