Not sure I agree with you on Southampton. Lambert is probably the one player that stands out, but he's a hard worker that would dependably put a shift in. Shcneidelin, Wanyama, Davies, Rodgriguez, Lallana all suited that style imo.
It's certainly a stand out problem. It's also (imo) noticeable that our pressing has seemingly gotten worse as the season has progressed. Somewhat worryingly I thought the same pattern was evident in both of AVB's seasons at the club.
I think one reason (not excuse) is that we have EL football. Means two games per week, not from a stamina point of view, but just from a time on the training ground working on stuff it might be a bit limiting compared to what Poch had at Southampton and other examples.
I do think some of our players aren't really suited to that style, at least not currently. Ade, Eriksen and Townsend come to mind. As I have mentioned in other threads I would like to see us start more players like the one's you mention in an attempt to up our pressing game over the coming games. A high pressing game really does depend on all the players working together, no room for passengers.