I agree every team that wins the cup is full of hard working players and near stars but they're typically anchored by at least 2 or 3 star players as well. Here are the teams that have won the cup in the last 15 years:
Colorado - MacKinnon, Makar, Rantanaan, Landiskog
Tampa Bay - Kucherov, Stamkos, Hedman
St. Louis - Tarasenko, Pietrangelo (I think this is the type of team you're describing)
Capitals - Ovechkin, Backstrom, Carlson
Penguins - Crosby, Malkin, Letang
Blackhawks - Kane, Toews, Keith
Kings - Kopitar, Doughty
Bruins - Chara, Thomas
Redwings - Lidstrom, Datsyuk, Zetterberg
Ducks - Getzlaf, Perry, Niedermayer, Pronger
Of those 15 teams, I think only a single team fits your description and that would be the one and done Blues. While star players are no guarantee of a cup, it would seem winning the cup without them is near impossible.