Xbox 360 didn't damage the "Final Fantasy" brand. There are a few reasons why.
Final Fantasy XIII was a radical change from the series norm. The game was also hyped to high hell by the fans. So smaller disappointments became massive ones due to how it was being held on such a pedestal.
Final Fantasy XIV had such a poor launch, and a very flawed design structure that I don't even know what the hell happened. Combined with a premature launch caused a catastrophe. Lucikly the new team has things under control, and plans to fix it with 2.0 later this year.
I don't think that the Final Fantasy brand has been damaged a lot, but the initial FFXIV launch was a massive disappointment. I just hope to high hell Square can deliver on Versus (oh and they can't either) because the fans are setting themselves up for failure yet again.
Square-Enix is not a perfect company. They make mistakes, they're humans just like the rest of us (I hope).