Yes they did - but that is because they both had crap goalkeepers. Not because they had a world-class goalkeeper on the line - who wasn't very good with the ball at his feet.
Pep had Joe Hart who at the time was regarded as one of the best in the league, probably second or third behind Cech and De Gea (who was at the peak of his powers around this from memory). Is De Gea currently 2nd or 3rd best keeper in the league?
Alisson and Ederson are the undoubted top 2 this season followed by Ramsdale (IMO based on this season at least).
Then you have Raya, Pope, Martinez and De Gea - probably in that order (again just my opinion based on this season).
So I would argue that Pep had a better keeper in Joe Hart when he joined compared to today’s De Gea which ETH has inherited.
I agree Klopp landed with a shitshow in goal with Mignolet. But even with his slim budget the following summer he still went out and bought a new keeper in his first window in Karius for £4.7m. Karius was more of the mould Klopp wanted, but cheaping out in that area obviously didn’t work out, so as soon as they got the Coutinho money, half of that went on Alisson as it was such a crucial area of the pitch to solve.
Alongside sorting out his defence by spending the other half on Van Dijk, Liverpool looked a different team and instantly started seriously challenging.
I believe we’ve largely sorted the defence now with Varane, Martinez and Shaw, plus maybe a new RB this summer, the last piece of the puzzle is a keeper that can not only make great saves, but commands his box, is quick off his line to support a higher line, is composed and secure on the ball to exude composure and confidence throughout the rest of the team.
Dave makes great saves, same as Hart used to offer. But even though Hart was commanding, he still lacked the composure on the ball Pep wanted to set his team up from the build up, so it was pretty much the first change Pep made.