Only if they are spending money on microtransactions while playing it, MS pays MLB for the show to be on gamepass, Sony isn't going to be putting their own games onto the service and people who play MLB on PS5 actually pay full price to play it lol.
I know the bungie games will come out on xbox they've already said that but it's going to be few and far between as far as anything else. You'll see a lot more xbox games going to PlayStation than the other way around. Xbox doesn't have anywhere near as many consoles sold and people don't buy enough games on the platform. PC yes you'll see all the GaaS titles day and date on PC, I still don't think they'll do the single player stuff day one simply because they want people to keep buying consoles. They may shorten the wait for bigger games but I don't believe it'll be day 1.