Talking of logic, it's logical to assume the capacity problems were in the authentication servers as these affect everyone, and also these weren't capacity tested when the service launched as there was no authentication on the first night.

I came to same conclusion.
BT's PR statement has a touch of arrogance about it, as if they are suggesting people who tried to login during the event were doing something wrong.
On the forums a few posted they got kicked off after logging in before the mtach and of course at that point they couldnt get back on again. So BT's statement is by no means accurate.
There was authentication prior to launch.
BT Infinity 2 Since Dec 2012 -
BQM