Ted Lasso wrote:He's not a Mourinho player because he cannot defend deep consistently. He is nothing like John Terry. He is more of a 6'3'' Mascherano. Not that that's a great comparison either.
Also, he makes plenty of errors in big games. Watch the Champions League final to see how many times Ashley Cole alone had to cover for him. Iirc, he actually yelled at Luiz at some point in the second half.
He makes two types of errors; positional ones when he has to withstand pressure and possession-related ones when he mocks around overconfidently with the ball. But he is good at one on ones defensively, he is fast, and he is very skilled so people think he is a world class defender.
He would be much better off on a team where his skills on the ball are more vital and he can defend much higher.
Like I said it's in terms of ability and talent, anything JT can do as a physical and technical attribute Luiz can do and some he can do much better. He does whatever Mourinho wants from a CB based on talent, he's always had these players, Carvalho-Terry-Lucio-Ramos-Varane these are guys with around the same physical and technical attributes and Mourinho has always wanted these attributes from his defenders. Again, talent wise. 
He's actually a really good player in the big games, he doesn't make that many errors in big games surprisingly, he does get really complacent a lot of the time. I did watch that game I still have it recorded on my TV, like 5DOM said Cole yells A LOT on any given game, that's his nature, he'll yell even if he's complimenting the guy and I do remember that, that was the only mistake Luiz made that whole game. 
That game was a prime example, despite them dominating the game the only good chance they could muster was the Robben chance at which Cech saved with his feet and the Gomez miss, other than that despite having the ball for 85 minutes they couldn't challenge Cech (not even from the penalty spot), that's all you can ask for a defence that was under pressure for the entire game plus extra time. Him and Cahill were put through the test that game not just because they came back from injury but they were always busy and they were magnificent. 
What you described was two errors defenders can make in general, every defender has been guilty of that. Heck one of my criticism of Cahill is that he's really slow to react but has good recovery, which is why he has plenty of these last minute tackles in his portfolio and most of the time he gets beat on the dribble are ones he could've easily prevented if he was either more aggressive or can diagnose a play quicker, but to his credit he's been really good for us this year. That could be seen as positional errors, also he is a bit cocky and arrogant but I think most teams game plan against Chelsea is not let that midfield have the ball, so he's forced to come out of the back and sometimes ping the ball, it's what to every big team does. Look at United when they had Rio, Barcelona with Pique, RM with Ramos, Hummels with BVB etc
He's actually not that fast, he's got good recovery speed but not great. He'd be terrible at a high line, he's good as long as he has someone covering him. Basically win the ball around the midfield area and initiate attacks quickly, that's why he's so valuable to us. When the opponent has the ball he's so strong and agile that he'll win the ball around the midfield and that exploits the opponent as they don't have the numbers to defend