Eh.
I didn't think so. There's only so much of "Friendship is Power" I can take before my suspension of disbelief is broken.
The ending of Awakening broke it into a million pieces...
"Grima has mortally wounded all of my friends in a flash and is now trying to control me, but I can overcome it and heal my friends because INVISIBLE TIES!!"
"By killing Grima, I am technically killing myself, but I survived anyway because FRIENDSHIP!"
Quick note- Robin doesn't magically heal everyone. Naga intervenes and heals them.
I'm okay with some hammyness, but I don't feel as if it reeked of ham. Fire Emblem: Awakening's biggest claim to fame was the interaction between all of the characters. Yeah, Robin's popular, but everyone has a favorite side character. Some love Owain's lunacy, others Frederick's mothering, and some Gaius' charm. The entire gameplay focus around these "Invisible Ties," or, Supports.
At least Robin didn't suddenly perform some superhuman feat such as flying or obliterating an army of Risen- all he simply did was break out of mind control. Yeah, the Big Bad was controlling him and it was kinda miraculous knowing Grima's strength, but as we've seen in many movies and games and even real life sometimes, people with strong enough hearts can pull strength from somewhere they've never known. Not to get into the "science" of mind control, but Robin possibly had an adrenaline rush and was able to push aside Grima's influence when he was touched by his friends' pleas. Also, Grima was substantially weakened by his travel to the past, so that's another possible contributing factor. Robin's friends didn't grant him some magical power- they just encouraged him to overcome mind control (something that didn't happen in the bad future where Robin was separated from the rest except for Chrom and was caught off-guard).
Concerning Robin surviving killing Grima, yes, he was killing himself, but that was him from another time. Yeah, there was some sort of psychic/mental connection between the the two, but killing a version of yourself from another dimension shouldn't make you cease to exist then and there. If baby Lucina grows up and murders the time travelling Lucina, the younger one won't just suddenly cease to exist. Grima is just a possible path Robin's life may have taken that he ended. Yeah, Robin did disappear, but it's possible that this disappearance was to kill the link that he shared with Grima. Or maybe Naga somehow saved him. His will was obviously connected with the world still, like ghosts seem to be in some media, so he remained attached to the world once the link with Grima was gone.
Oops, I went and wrote too much again. I just wanted to say that while friendship did influence some of the events in FE:A, it wasn't overly hammy or even a Deus Ex Machina- it was motivation to drive Robin.