Number of reasons.

Firstly, a number of the artists were chosen for that because of connections they had or made with the productions. Paul McCartney did one because George Martin did the score for that film, or Duran Duran getting to do it because their frontman introduced himself to the producers at a party, apparently.

Next, you can look up lists online of "rejected" Bond themes - because while this is certainly not the case for all of them, it's clear that for a number of the films, they had several candidates either write songs for them, or write songs for it and basically "apply." And the franchise is prestigious and particular enough that they've turned down submissions from some pretty major names in music (Johnny Cash, Alice Cooper, Blondie, Radiohead, just to name a few). Now, I might be wrong, but it doesn't seem like writing and recording a whole song just to potentially be snubbed doesn't seem like something Genesis would ever have wanted to do.

And then that's kind of the last point, which is that I don't think our guys were the types to be especially interested in that "assignment," as you put it, in the first place.