It is unfortunate that the Nazis chose to pollute such a peaceful and historical symbol. The swastika is NOT an 'evil' symbol, and it has a long and interesting history. It is - in fact - quite 'cool'.
Believe it or not? I'm going to choose 'not'.
There are billions of people in the Indo-Asia area which believe strongly that this is a valid, and respected religious symbol. To me, that does not equal his belief being 'normal and commonly held'. Maybe among ignorant fools.
It is a shame that the Nazis did that, but it's done and there's no going back. In the US in particular the swastika is almost always associated with the Nazis simply because WWII and the Holocaust are rather major historical events. Not to mention there's plenty of aryan brotherhood groups still using the swastika as a symbol to promote everything wrong the Nazis did. It's very much still in use as a symbol of hate.
Believe it? Sure I do, I've known about it for a long time, doesn't change the fact that most people in the US (also Europe, Canada) associate it with the Nazis. And your argument kinda runs off the rails in that bolded part. We're not dealing with the Indo-Asia area. We're dealing with North America. Even if you try to argue that people from that area use Xbox Live, Microsoft is still a US company. Their policies on acceptable use are going to by and large reflect that.
Personally I've yet to see someone use the swastika online that was trying to promote discussion about it. Every time, without exception, I've encountered it on forums and such it was someone trying to promote the "ideals" of Nazi Germany and/or harassing people. So yeah, people who use it online by and large are doing so to promote the ideas of the Nazis. I'm certain there are exceptions, but they're few enough that it makes sense for site owners to ban it. Besides, if your goal is education, using it as an avatar is a bad idea. Most people would simply see it and never ask you why you used it. Hardly a teaching moment there.
Besides, this is a total non-issue. Xbox Live's Terms of Service forbids this type of thing. You agreed to that ToS when you signed up. So it's not so much about the swastika as it is about violating a contract. (ToS's are a contract between the user and the company.) And really, online games are NOT the place to even try to educate people about this issue. When you're playing a game you want to have fun, to escape reality, not to get a 5 minute lecture on why the swastika's actually a totally cool symbol that the Nazis corrupted.