Xbox 360 connects to internet but, not Xbox Live...

Hello,

I've been unable to connect to Xbox Live through my Xbox 360. The console
connects to the internet with no issues (other than not supporting WPA2 encryption, meaning I had to disable the security/password, which is far from ideal), I'm able to view the available add-ons, apps, etc. The console even downloaded two (supposedly)
mandatory updates, one for Xbox Live, one for the main Xbox home screen. However, when I try to download/use any of the available DLC, I'm told I need an Xbox Live account... which I already have.

Under the 'social' tab, it shows I'm "SIGNED IN". Whenever I choose the "sign in" option when prompted, it takes me to the animated avatar, which also shows as "SIGNED IN".

I've ran the Xbox Live Connection Test, it shows that all points are "connected", claiming that I CAN connect to Xbox Live, the Live Status points below it all show as "up" (as in online, connected, functioning, etc). The Network Statistic
tests also show no issues. However, when I try to connect to Xbox Live through the big black icon on the home screen (or even the "join the fun" link on the 'social' tab), it says "We can't access Xbox Live through your console".

So, what's going on? Is Xbox Live now discontinued for Xbox 360 owners?

Just to clarify, I'm using my phones internet on the Xbox, so there's no modem/router to mess about with. That said,
the internet connection itself is obviously not the issue, it's specifically the Xbox Live service that's causing problems here.

:)