Montreal should be given a lot of credit for doing so well in the playoffs last year, and even the year prior. So yes - that should count. Should it count enough for them to be voted here, or soon? Well - no. I get there's a ton of question marks for Montreal's roster, so I'm not here trying to advocate for my team to be voted in now. But their recent playoff success does count, yes. And it should probably gain them at least 5-10 ranks in the final votes vs where they would have been had they been out in round 1 last year. Doesn't mean they'll be voted in 4th though, probably closer to ~12th to 15th I predict.
San Jose has sucked for a while. New Jersey too. They're bad examples. Nashville has been trending downwards for a while, also bad example.
I think recent success should absolutely matter, as it gives credence to how consistent you expect a team/roster to be in future.
Florida - hasn't won a playoff series in the past 4 years (when I say past 25, it's just to mock them - past 4 years though, 100% relevant to discussion at hand). They only made the playoffs once in past 4 years. Does that inspire enough confidence to say they definitely make the playoffs this year and are a top contender to win it all? Nope.
Teams with recent playoff success - or more consistent regular season success - should be voted above them imo. Teams like Pittsburgh, Washington, Boston, Carolina, etc. Even Dallas in the west made the finals 2 years ago, and lots of playoff rounds won in recent years, or Blues who won cup 3 years ago. I'm not saying all those teams necessarily have to be ahead of Florida - but many of them should.
Until Florida proves me wrong and manages to qualify for the playoffs again, and mostly show some level of playoff success - I simply don't buy them as a credible threat to win the cup.