Yeah, medical staff in hockey(Sports in general for that matter) work a hell of a lot different than your typical medical protocol.
Anytime a player is injured in hockey and they go to the locker room, first thing they look at is whether it's possible to get the guy back on the ice. If its a cut to the face, broken nose, even things like minor sprains or separated shoulders... The first thing they are doing is seeing what if anything, can be done to get the player back into the game. Especially in the playoffs. And the players are pushing that same mind set. First thing the players almost always ask is if they can get back into the game.
So I dont think the medical staff deserves any blame at all for Landy. Guaranteed they went in, tried to patch up the cut, stop the bleeding or control the bleeding and then had Landy go put some weight on it give it a test run to see if he could go on it. Obviously he couldn't go, and likely after the game when they looked at it again it had probably swollen up or started bleeding again, or they were able to take a better look at it and saw that it was deeper then originally expected and that's when they realized it was more serious then originally thought.
But ya, in game like that 100% they're going to do what they can to get him back on the ice and Landeskog would be in that exact same mind set.
When I saw a couple games in Calgary last year, they had a feature in the pregame about the Calgary medical staff and that was what they were talking about, any time a player came into the dressing room with injuries the first thing they did was assess whether they could get the player back in the game.