It's not a case of what the other team wants. They're trying to win the game and I'm sure they couldn't care less if Ovi scores an EN goal. I just don't like seeing him hanging around the blueline hoping for a puck to break free while the other four guys are trying to stop the opponent from scoring.
This is actually a fairly complex issue.
I would remind you that Ovie's ENG previous to last night's occurred when he blocked a point shot and the puck bounded out to center ice, where he was able to pick it up. Forwards tend to play higher positions on defense and the defensemen (and sometimes the center) the lower positions. So Ovie, as with a lot of forwards, has better chances to get out on ENG situations, with the opponents looking to press down toward the goal. And he isn't ALWAYS just cherry-picking.
There are other factors to take into account. Teams now are being more aggressive in pulling their goalies -- pulling them with several minutes remaining on the clock instead of just within the last two minutes, and, because of that, pulling a goalie again if a team scores an ENG. This is important in Ovie's case, because Carbery seems to minimize using him when the goalie is pulled with his team up just one, Ovie being a relative defensive liability. But if the Caps score and the opponent pulls the goalie AGAIN, then the pressure is off and Ovie is out there more. Ovie can't help this -- if a team spends more minutes with their goal empty, then Ovie needs to be out there at least some of the time. It isn't like when a team pulls the goalie with a minute or 90 seconds left and the team with the lead will use its best defensive players exclusively, with the end of the game close at hand.
I agree with you - it doesn't look good or feel quite right. But it is what it is, and it is at least partially because of the strategy in the modern game, so it wouldn't be a surprise to see more of it. I'm hoping, though, that at least the goals to tie and pass Gretzky, if/when they occur, come in the course of non-ENG play.