No single 3D API is capable of supporting all 3 of those platforms. However, if those are the only 3 you're concerned with, then Direct3D gets you farther (Windows+Xbox) than Vulkan (Windows only). You're still going to have to write/use something different for PlayStation (which has its own custom API).
The two major platforms that favor Vulkan are Android and Switch, but the former is in the "mobile" category rather than the "desktop/console" category and few games overlap both categories.
But really, Asahi Linux on M1 Mac is about as niche as you can get, so broad hardware support for native APIs isn't really relevant, and I don't think AAA games are necessarily what's meant to be supported either, since they generally won't work for other reasons (can't install x86-64 Windows kernel driver DRM/anticheat on an ARM Linux machine).
AAA gaming is THE driving factor for getting Vulkan on Asahi. There really are only a tiny subset of games that require a kernel anti cheat. Let alone the wealth of single player games.
I'd love to hear about all these DRM-free single-player AAA games. If you're talking about older games, then maybe we're just talking past each other; I assumed we were talking about recent AAA games.
Recent AAA games absolutely run on Linux using Proton. DRM isn't really a concern. The biggest and most successful DRM vendor, Denuvo, even goes out of their way to make sure their product does not interfere with Proton compatibility.
It's pretty rare for a new game to come out and not be playable on the Steam Deck within a week if not on day 1. And when that does happen, it's usually a PvP multiplayer game with kernel-level anticheat.
I'd assume Denuvo DRM works on Linux x86-64 because the company has made it work on that platform. I'd be surprised, with FEX in the middle, if it doesn't get bent out of shape running on ARM: either flagging itself as having been tampered with, or the game. Maybe the company also has a solution for ARM support though.
I do think the question of "why Vulkan" has been thoroughly answered though: 1. DXVK means Direct3D->Vulkan translation already exists (but not the other direction) and 2. Proton already proves that Vulkan gets you most AAA games (with the exceptions having nothing to do with the 3D API).
