It looks more like how to avoid paying Windows licenses for the SteamDeck to me.
That's a small part of it, I think. They've almost certainly spent a lot more on pouring time and effort into Linux than they ever would have saved on license fees. It seems like Valve doesn't want to be beholden to Microsoft in any way. They support Windows because that's where the users and the games are, but they don't want Microsoft to be able to rug-pull them either.
I don't remember how officially it was stated, but original push by Valve for steam on Linux with Proton was to remove their dependency on Microsoft - a hedge against possible future ecosystem-impacting decisions in Redmond.
Making SteamDeck use windows wouldn't impact prices much, Microsoft is really friendly for putting windows by OEMs. Could even run modified to act like current steam deck.
Instead, SteamDeck is there to drive up testing on Proton or straight forward porting to Linux, which just availability on Linux and the previous steam machine didn't drive up
Close, but the root is more nuanced. Once upon a time, Microsoft was talking about regulating "apps" on windows like Apple does for the iPhone. Valve saw the writing on the wall: a potential ban on violent or otherwise adult-themed games. So Valve started the steambox project. Get the games running on linux/WINE and they could tell Microsoft to push off. Years later, we have the steamdeck as a revolutionary product and linux is the go-to OS for portable gaming.
Linux/Unix has been used as a base overwhelmingly in pretty much every new consumer OS for decades. Not to mention Microsoft certainly cuts deals with manufacturers when it comes to windows on portable devices (I think at one point they offered free licenses on devices with screen sizes under 8 inches).
The steam deck is 100% usable without leaving 'game mode' even a single time. Something that is genuinely impossible using Windows as a base. That's the important part
The amount of money Valve has pumped into Linux would have far exceeded the money they saved through licensing Windows. Like probably by an order of magnitude or more. For someone as smart as you seem to be, your points don't make a whole lot of sense.
I'd guess SteamOS is more about control than licensing costs.
I don't think Valve wants to be at the mercy of Microsoft and their policy & technical decisions.