If Xabi doesn’t choose us, my other choice would be Simone Inzaghi. He is one of the most promising coaches of the new generation, an excellent cup manager with Coppa and Supercoppa Italia wins for almost every season of his managerial career so far (with 2 different teams: Lazio and Inter) plus obviously taking Inter to the CL final for the first time in 13 years, where they were rather unlucky to lose to City after mostly outplaying them.
Like Xabi he plays 3-5-2 and his Inter team is currently head and shoulders above the competition in Serie A. He doesn’t have as clear a style of football as Xabi, Arteta or Pep; he seems to be from a more adaptable school like Ancelotti; his man-management skills and ability to get the most out of his players have won repeated praise. His teams can switch between high press and mid-block, they vary their attacking build-up so it’s difficult to counteract and if you look at Inter this season, the players play with joy and freedom, in addition to being obviously excellently organized.
I have no idea if he will be interested in leaving his current job and coaching Liverpool (his contract runs until summer 2025), but if he is, he should be ahead of Xabi or Amorim purely based on experience and history of success. It would feel a little like stepping into unknown to hire someone who doesn’t have a prior connection with Liverpool (if you don’t count being coached by LFC fan Sven-Goran Eriksson), but to be fair neither did Klopp. For sure I would avoid German managers for now - this would feel like we’re trying to replace Klopp and that would be a mistake. Just like in recruiting players, the best strategy is to forget the hype and extraneous factors and look at underlying stats to select the outstanding candidate, no matter where they come from.