I think the things that hampered FFXIII are not specifically to do with the name Final Fantasy as much as the expectations placed on the JRPG format itself thanks in large part to the success of Final Fantasy since the Playstation.
It has become de rigeur for a JRPG story to confuse pretention for grandiosity, because dammit it worked for FFVII (Let's face it, the story of Jenova is just the story of Lavos from Chrono Trigger with a bunch of unnecessary cruft about lifestreams and whatnot), there are very few that actually connect us to the protagonist on a personal level like, say, Breath of Fire 3 did.
The battle system, however, is one of the best in any JRPG, and certainly the best in a Final Fantasy. It makes the most of the fact that there is no attrition to make almost every combat a meaningful challenge, usually with a solution based not on luck or brute force of levels but on understanding and manipulating the system provided to you, using the right mix of classes at the right time, and changing them in response to the flow of the battle. You might not be clicking on "attack" every round yourself, but you'll be making far more decisions than you would in any other FF game.
The problem with it is repetition, because there are quite a low number of potential encounter groups in each area the player will find themself fighting the same encounter repeatedly, and because the nature of the encounters is that they are now almost a puzzle, when you have solved the puzzle there is no need to modify your approach. Contrast that with, say, Enchanted Arms, an otherwise pretty terrible game which had a similar restoration of characters between battles, but which rationed the restoration of HP and MP (used by all attacks) by the reduction of another resource, so there was always an incentive to improve your solution to the same encounters so that you could keep fighting further towards the next save point.
Had FFXIII had either a system like that to push the player to continue thinking even in repeated encounters to refine their approach to them, or altered the encounter design to provide a steadily staged and increasing challenge curve through each area (a tricky task), it would have fully succeeded in what it attempted to do.