Ace Combat 5 has the best produced story, and the animation is excellent. It looks like a semi-hollywood cgi movie in the cutscenes.
But AC5 also had a cheesy storyline. That song that they kept playing over and over again, the one the dude who died liked and the people sang in the stadium, was about the stupidest song anyone could have written, it had about 3 notes and was the same line over and over again.
I haven't seen ac6, cause I don't do arcade air combat any more, but I know Ace Combat 4 had a far better story than AC5. It had only half as many missions, but the story was first-rate; all about the enemy ace's human side, and then at the end you have to kill him.
All Ace Combat games allow you to pull about 20g's in turns at or near mach speed which is ridiculous. And have lots of su-37's and 47's in the air at once which is even more ridiculous since there's only two of each. So the AC with the best story wins.