What exactly does Luiz need to change or improve now? He's perfect for that central CB role. He doesn't need to change or improve on anything but his concentration and consistency which has nothing to do with his raw footballing ability.
Still Luiz is being doubted on his one on one ability, when not one of the top players or strikers in the PL have got the better of him. We've played against Liverpool, Arsenal, Spurs, Everton, City and United, the teams with the best forwards in the league, and apart from against Arsenal and Liverpool performed very well against all of them. Even against Arsenal and Liverpool he didn't look out of place - he certainly wasn't the worst player or defender on the pitch. So Luiz has faced the likes of Aguero, Kane, Ibrahimovic, Lukaku etc. and didn't allow one mistake that lead to even a goal scoring chance or goal but he's still essentially being labelled as a liability in the back, as someone you can't fully trust yet which is quite incredible.
Bonucci was the same as Luiz before Conte was done with him, so it's only fair that you give Luiz the same time and luxury to work under Conte and prove himself which so far he has done well. He's been one of the best CBs in the league period.
I think the LCB role is a lot different to the LB role that Azpi played thus you can't really compare the situation. Our LCB (Cahill) does not have to bomb up the pitch and deliver a cross or check back on to his right foot to deliver a cross. When we have the ball they are important in keeping the ball moving, which is why I think Christensen would do just fine their to alleviate your concerns. How often do you see Azpi or Cahill even in the 2nd third of the pitch? Not often at all
Christensen is a passer, he's comfortable enough with his weaker foot to pass it forward well if he needs to. As we've seen with Cahill in the system however it can work. Now imagine we have someone who is far more comfortable on the ball and someone who's a passer and good at moving the ball forward. It would add another dynamic to our forward play. Christensen for me would be amazing in Cahill's role.