Otherwise, we could say something like "COVID outcompetes the flu".
I don't think we have the data on Omicron vs Delta yet as you say.
I mean, it does, but ... we don't really know that Omicron is outcompeting Delta in that sense. We know that Omicron case counts are growing, relative to Delta, but we don't know how well Omicron prevents later infection by Delta, etc.
You asked "Wouldn't the fact that it's outcompeting Delta imply that there is some cross immunity?" The only fact that we have right now is that Omicron is growing quicker than Delta, not that Omicron is somehow suppressing Delta with cross-immunity.
> Otherwise, we could say something like "COVID outcompetes the flu".
Well--- COVID causing everyone to be careful, combined with influenza's lower Rt due to not being novel, really put a dent in influenza. So even then, without cross-immunity, you can have growth in one suppress growth in the other.