I know you've been involved with cricket to a good level, not sure if you have with football too, but I'll go on the basis you have. What he asks them to do, is very very high risk. Every player to push up the pitch, when all your instincts say you won't be good enough to sustain pressure on the ball is really difficult for players. What you want to do, especially down to 10 at OT, is just drop deeper, and have them play in front of you.
I know Klopp is very much in vogue, but he has a wage bill of £300m+, maybe even £400m now. It is likely 2-3 times that of Southampton. They have the players who can do it successfully.
If you are Southampton as a fan, you kind of have to accept that on some occasions, when it goes wrong, it's going to go badly wrong. If you can accept that, there will be good performances.
Mentally though, it will be hard. Going into the next game, he will be saying to them to get up the pitch and press. If 1 player has doubts, and doesn't fancy doing it, the chain will break.