Actually they don't even need their own shooter. With this move, most of the sales will be on PS. This means more MTX sales on PS and PS gets their cut. Without spending money, PS gets massive money.
Okay that's great for SIE, but what about PlayStation customers?
If alongside COD there are ever say 4 other AAA games coming out in a given year that are going to be on both platforms, but all 4 would also be in Game Pass Day 1, why would I pay $350 for all 5 separately on PlayStation if I could get all five in a single year's Game Pass sub for $240? Especially if these are games I'm only maybe going to play once through, maybe a second playthrough for some within that same year?
This always seemed like what Microsoft was banking on to make Game Pass take off: appealing to core gamers who aren't necessarily frugal with spending, but want to optimize their spending dollar, and plays a lot of big (and some small) games a year.
In practice MS's output cadence isn't good enough to realize 4-5 AAA releases a year (they can't depend on 3P to put AAA games in GP Day 1, for the most part). So for now yeah, most core gamers who know those 4 or so AAA bangers are coming to PlayStation in a given year (most of which won't be in Game Pass Day 1 or within 3/6/12 months from release) still make out good here and be better off buying on PlayStation assuming they want to play a game like COD online year-round.
But at some point that could start looking different, especially if SIE don't introduce any other means of accessing games on their platform (i.e like the per-game subscription/pay installment for PS Store model I had talked about a while back). They don't want to get caught in a situation where something like Game Pass actually does start delivering consistent quality of big AAA and smaller AA/indie games at a price that starts looking really appealing vs. buying those games each Day 1 at $70 or so bucks on PlayStation.
This is even more a potential sour point considering SIE's 1P (aside from Insomniac) haven't necessarily been delivering a ton of 1P releases annually so far, SIE have been making a (IMO) bad move porting virtually all of their current-gen games to PC (well before the PS5 Pro even launches), and the potential for certain 3P timed exclusives might be dwindling (future mainline FF games, for example).
Like, I've seen people joking that MS going 3P are bringing more games to PlayStation than SIE studios are. Well, this is the potential downside (for SIE) in that scenario, if indeed Xbox as a console is dead/dying (or at least very weak) but Game Pass as a service lives on. Who have been the ones saying it's a "platform war" now, btw?