On Criterion's site, they just posted a somewhat long explanation of why they couldn't bring Burnout Paradise online play to Xbox 360 owners who don't have the optional hard drive. Essentially, it boiled down to preserving the game or preserving some of the audience. And Criterion chose the game.

We opened a dialogue with Microsoft to attempt to discover a possible solution and all possible technical options were investigated to retain the seamless freedom of Burnout without requiring the HDD. Unfortunately a solution was not available and we decided together that it would be in the best interests of the Burnout community if we require the HDD for full online play.

Psst, Criterion. Allow me to let you in on a little secret: nobody cares about Xbox Arcade owners. The apology is nice, but it's the equivalent of saying you're sorry to a a guy who is too stupid to stop punching himself in the balls, even though it hurts like hell every time.

XBOX 360 HDD/ONLINE STATEMENT [Criterion via N4G]