@Robby:
They'd already stated pretty explicitly for two movies that Azog was off and about trying to kill Thorin because he wanted to and his master wanted him to… and sending his son to fill in when he was called off... and similarly that Thorin was all uppity to go take him on personally for similar reasons. So the personal angle was there... between those two... but no that wasn't going to singlehandedly be the reason for the full commitment of forces.
I am specifically referring to the third film, where I don't think we even got the token "hated enemy! I have waited ages to avenge such and such" spiel. The final battle was more a case of "we need to take out the Boss Orc to stop the fight!" and less about personal feelings. And from Azogs perspective, they pushed the angle of re-forming Saurons kingdom…but at a moment in the film after Sauron was banished. Azog is acting on behalf of a spirit that just got trounced by Galadriel! (yes I know he didn't know that, but we knew)
Azog and Thorins relationship was put on the back burner in the third film, much like how film 2 made a big deal about Bard being the descendant of the guy who tried and failed to kill Smaug the first time around, and film 3 lets this dramatic plot point go unused. The 4 hour Hobbit movie condensations are already popping up, which propably alleviates this a bit... but when you chop your story into 3 bits you can't just leave all the character relationships in the first two- even if your 3rd film is an entire 3rd act. It'd be like if film 3 had no scenes of Bilbo and Thorin interacting, and then expected us to get invested in the end.
@Robby:
Helm's Deep didn't have much higher stakes beyond "the orcs are attacking, gotta defend cause otherwise a lot of innocent people die" and the battle with the ents was "take out Sarumon's strategically valuable location and stop a specific enemy general". And technically, Sarumon wasn't even really connected to Sauron. As a franchise it had a lot of battles like that.
Helms Deep wasn't the big final story-closing trilogy capper though. And even so, its still easier to get emotionally invested in "Sauron wants all humanity dead, so here hes trying to kill a shitload of humans" as opposed to him just wanting the keep, because it'd give him an advantage in Middle Earth risk.
@Robby:
But yes,as the final battle, it SHOULD have been bigger. Moments for all the dwarves (at least one unique final memorable attack each) and a better view of how it ended beyond "there are eagles!".
Honestly, I think the "epification" is the biggest misstep of these films. They wanted the finale to be big and grandiose and a "defining moment!", but couldn't quite back it up while still adhering to the source. This is just off the top of my mid, I'd have found it more convincing if, say, Smaug had merely been repelled initially, and then agreed to lend his services to the orcs if they teamed up to kill everyone and get him his mountain back. And also have the Dol Guldur battle coincide with the Erebor battle instead of preceding it. Idunno, It'd feel like there was more at stake.
So either go all-out, or scale down the epicness if you can't support it.