> Doesn't the fact that it's outcompeting Delta imply that there is some cross immunity?

Not really. If Delta's Rt is greatly reduced because lots of people have immunity, Omicron can show up with a much higher Rt and grow in case numbers. It doesn't need to further reduce Delta's Rt to "win".

I.e. even if R0 among those naive to the variants is similar, once you have 2/3rds of people who can't get Delta because of vaccination or prior infection and that can get Omicron, ... Omicron is going to win. And it looks like Omicron may be more contagious than Delta at baseline, so it's even quicker.