Yes, he came through, even if it took the dad to pull him out of the misery and his very own daughter's reminder about his own wise words, to complete. He was his father's son all along and all the way, his mind had just been poisoned for too long and too bitterly to heal and turn him into the perfect king just like that.

I think two lines in the part 2 put it into nutshell.
SIMBA: "I just need to follow in my father's footsteps." (or something like that.)
KIARA (crying): "You'll never be Mufasa!"
And that clearly affected Simba immediately.

He wanted to and did live up to everything, just had many troubles and made mistakes for very understandable, psychological reasons, whereas he could've easily enough made some anyways, seeing to the prouod and reckless prince he was.

He did live to his promise :D

he is a great king but he "will never be mufasa!"

I believe so, but it was a bumpy road. In the first film alone, we don't really get to see what he is like as a king - we just know that he has finally taken his rightful place as king. However, in the sequel, it is shown that he struggles to overcome his distrust for the outlanders.

Fortunately for him, his daughter appears to have inherited some of Mufasa's wisdom! I do think that Simba has Mufasa's wisdom in him too, but it's kind of dormant in him for some time and he just needs a little help from his daughter to think straight about some of his decisions (like including the outlanders). He ends up being a great king who gets more than his fair share of challenges thrown at him. He has his flaws but his character develops wonderfully throughout the films and he gets there in the end. :)