I think you're correct - if Omicron and Delta provided no cross-immunity, there would be no competition, and they would just continue on their own trajectories

I think you're correct - if Omicron and Delta provided no cross-immunity, there would be no competition, and they would just continue on their own trajectories as if the other didn't exist. But, I think that's not actually the case. Omicron infection does provide some immunity against Delta. See this preliminary report from South Africa, for example: https://www.ahri.org/omicron-infection-enhances-neutralising-immunity-against-delta/ https://www.ahri.org/omicron-infection-enhances-neutralising... I'm not an expert (so take this next part with a very large grain of salt), but I would guess that someone with an active Omicron infection (and so swarming with antibodies) would get the biggest immunity boost against Delta. So if Omicron is everywhere, Delta runs out of infectable hosts. In principle, it could even go extinct - if that were to happen, it wouldn't matter so much if Omicron-induced immunity wanes, there would be no Delta left once the shields are down.