Yeah I'm pretty much in the same boat since the update my Xbox Live disconnects automatically every time I start the game making playing it obviously impossible. If I exit the game my connection instantly is restored and other games work fine so I know its something purely related to Destiny. I mean damn not even Dice screwed up things this badly with Battlefield 4 and I thought that was the worst I was gonna see from a game for a while, I guess Bungie wanted first place in the screw department. What sort of pisses me off is the game worked fine I've only suffered session disconnection 3-6 times in the past 2 days all and all it was working perfectly for me. I have half a mind to deleted the update and see if that solves my issue since its arrival seemed to signal this current mess.
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I had the same problem (started after I had to download a 2mb update earlier) [b]Clear your system cache[/b] http://support.xbox.com/en-US/xbox-360/system/clear-system-cache then when you start destiny again, it will ask you to download a 2mb update. It's working fine for me, now.