I agree with you. It is not the fault of Klopp but the players because the allowed Man United to have an upper hand in the extra time which led to United pressuring their defense. However, it is not a must that Liverpool should win the game, and this is why I would say that luck didn't favor any team, but what favored Manchester united was the zeal and determination that they can still win the match. It was a good day for the red devils, as they were able to play with the spirit of the old Manchester united in this game, turning things around even when everybody thought that it is over for them.
Liverpool played not too bad but it is not their day and Manchester United players suddenly played better than they often shown recent weeks. I see this defeat is helpful for Liverpool because they will no longer have to worry about FA Cup. By that, they will have more chances to focus on Premier League and Europa League and winning either Premier League or Europa League will be bigger achievement for them than winning FA Cup.
Manchester United will have chance to win FA Cup to avoid another season without title. If they win it, it is a small achievement for them but it won't be enough to make their club strength is stronger and become more competitive next season. To be stronger, they have many things to do for improvement in their squad and tactics.