It seems to be a bit of a hallmark of Pep / "the City Way" to say things like "trust in the process" or "we focus on trying to play the way we want to play and the results will fall out of that". I don't know for sure that the players interpret it this way, but to me this means "ignore the occasion and don't let it get to your head". The Klopp approach appears to be the polar opposite - "feel the occasion and harness the energy". For knock-out competitions, Klopp's approach appears to be the superior one - if only because it may well be impossible to avoid letting occasions like this get to one's head. If you can't block it out, you may as well relish it. I love the rigor of Pep's football, but it can come off a little cold in these situations. There's a lot to admire about how Liverpool approaches it. But relying too much on this energy may explain somewhat lackluster performances against poorer opposition.
tl;dr - I can't help but feel that we're seeing a league-optimized approach vs. a cup-optimized approach.
I was pretty frustrated with Sane, but I will give him a bit of a pass because of the tactical adjustment that Pep made in order to give De Bruyne a free role and to allow him to drop deeper to provide extra options to beat the Liverpool press. This essentially meant that they were less likely to lose the ball between D and M, but had more trouble getting the ball to stick between M and F. Since there was no right wing option, the play up the channel to Sane became really predictable and easier to defend. Even after Sterling came on, I think they only used that outlet 2 or 3 times in ~20 minutes, perhaps because the pattern of play was too well-established.
For all the talk of sticking to the way they play, I feel like Pep's tinkering may have cost City dearly today.