Do teams tell players to be tops analytically? No.
But a team that takes an analytics based approach attempts to acquire players who have good puck possession numbers but are being undervalued by their team, such pittsburgh trading for Trevor Daley and Justin Schultz... two defencemen who are very good at moving the puck... or getting Hagelin, and Bonino, two guys who were good analytically.
Funny thing is teams that are top 5 analytically keep winning the cup.... see Chicago and LA...
And yes, if you are a good puck possession team, and make very few changes to your roster, you will probably still be a good puck possession team the following year (understanding that your roster will be a year older so young kids get better, veterans may regress etc...); but yes good possession teams generally stay good possession teams.
Does it guarantee a cup... no, some luck goes into any cup, injuries, a bounce here, a hot goalie, etc... but if you are near the tops you at least make yourself a contender. You can't guarantee a cup but you give yourself a better chance. The better regular season corsi team wins 70% of playoff series, don't you think that is a good thing to strive to achieve?
Since Corsi was tracked a top 5 team wins the cup every year, except one... the 2009 Penguins, who were a bad corsi team for half a season with Therrien coaching and no Sergei Gonchar, but were a top 3 team in the second half with Bylsma coaching and Gonchar back.
So