I think the team accomplishments should be 0% of the players evaluation.
However, I think that when you list off the reasons why a specific player was so vital to the team accomplishments then you're already making a list of that players individual accomplishments, which is what we should be doing the whole time.
In other words, saying "Clarke won 2 cups" should NOT be an argument but instead "when Clarke won those two cups he scored this amount of points, played this quality of defense, and also did x/y/z that helped the team win". That would be a fair argument because it's listing his individual actions/results.
Team accomplishments matter because there is merit in performing at a high level deep into a playoff run.
If you never make it past the second round of the playoffs, it's difficult to see how you perform under the hardest conditions possible for an NHL player. Now, it's not that you judge a player for not getting there (see Alex Ovechkin pre-2018), but you can judge them for how they perform when the get there.
Bernie Parent is a legend for winning two Conn Smythe trophies and the judgment comes from how he performed specifically in the Final in those Cup runs. You review his career differently if the Flyers lose to the Rangers in 74 or the Isles in 75.
I get saying that winning the two Cups alone shouldn't be part of the evaluation, but when people say Player X is an all time great and include how many Cups he won, it is implied that the player actually played a significant role in making that happen.
When talking about Parent, you often hear about the two Cups being an accomplishment, but you never hear about guys like, say, Craig Adams using two Cups to measure his individual success.
So, basically I agree with you, in principle, but the players have to get to a stage where you can measure them in the most difficult circumstances they can face as hockey players. That requires team success. And guys who get there with their teams multiple times will have a bigger sample size to judge them in those conditions.
Top players are vital to team success and you rarely have team success without the best players executing to get their team to those levels.