How many players care? How much pride would Kopitar feel putting on Team Europe colours?
On the record, every player cares because NHL is trying to hype it up as the most important tournament (until the next one) and supposely a replacement for the Olympics (plus im guessing they get more revenue out of this so it gives them every reason to promote this and not have their players go play in the Olympics).
Also all players are competitive so they'll likely treat this as something more than a pre-season game but fact is they likely aren't in ideal condition or at least as sharp as they could be heading into the tournament. This isn't taking place during/at the end of the season, this is taking place before the start of pre-season so players (despite working out all off-season) likely aren't in "game shape" so there's probably going to be a lot of sloppy play. Add to that not regular line-mates so the chemistry probably isn't great. The addition of team American (and to a degree team Europe) also can make things confusing for fans (as in which team to cheer for when its Team Can vs Team NA? lol).
Honestly if this tournament started say in Feb/Mar, i would probably watch every game but the timing of it (given its start of NFL season/end of MLB season), its just not as attractive as either and thats pretty much the competition (in terms of viewership).