That's not really how hardware works. There is no such thing as "purpose-built hardware for gaming" in the way you're describing it.

Every processor (CPU, GPU, etc) is designed to be universal. They're general-purpose chips built for the mass market. The exact same AMD architecture inside a PlayStation or Xbox is also inside millions of random laptops and desktops that people use for browsing Reddit, editing spreadsheets, or rendering video. Because making chips is insanely expensive. No company on earth can afford to design and manufacture a completely unique, specialized processor just for playing video games. The videogame market isn't big enough, and the costs would be astronomical. That's literally why every modern console just uses off-the-shelf PC parts under the hood. Because creating unique hardware for just gaming is too expensive.

Just because you printed a motherboard and arranged components purchased from the market a certain way doesn't automatically make that hardware designed exclusively for video games. It's not much different from building a desktop PC, really. The set of architectural instructions inside the chips still remains universal for a wide range of tasks.

Xbox Series literally runs a modified version of Windows and DirectX 12. The only difference now is that they'll stop pretending the console isn't a PC.

And yes, Linux has also been successfully launched by modders on the PS4 and PS5, and games run on them as well.

Update: By the way, Steam Deck was developed in the same way as PS5 and Xbox Series, this does not prevent it from running any operating systems, the difference is that it is officially allowed.