Having integrity in trying to win also doesn't involve playing Ritchie, Nordstrom, Nesterov, etc. It really feels like the opposite where they are directly hurting young talent by not playing them. The aforementioned guys have had 40+ games to show what they can do. Sure, they are fighting for a new contract but so are Mackey, Kylington, Simon, and Gawdin, and Phillips, Ruzicka, and Valimaki are all fighting to show they belong as regulars/belong in the NHL at all.
Markstrom playing is meh. He's injury prone but he's been hot and probably wants to work through things in game while he can, but again, Domingue has had zero opportunity to earn another contract and a previously injured goalie is going to be run for... some reason.
The lines they practiced with today aren't ideal, they are at least playing new faces in Simon and Ruzicka and sitting Nordstrom and Nesterov. Unfortunately Kylington is also sitting (Stone has had played well but this is where showing faith in players and letting them show what they can do is more integrity than running the same guy out there with no need for winning). At least the new lineup is something that could be a sign of change for future.
Also, wanting to lose is alright, and the "winning-culture" under other coaches has been questionable, but if Sutter is a genuine long-term option winning 2-3 extra games isn't the end of the world if it turns into results next year.