Ok, I didn't downvote you, but...
Its a Steam Controller.
For Steam.
... If you just made up some expectation, contrary to everything that Valve has said about this thing, and marketed it as...
To a certain extent, that is magical thinking, that is abandoning any concept of checking your hopes or expectations against... reality.
If you... want to run... a game... with this controller...
You install Steam, and the game.
Steam is the drivers, for the controller.
If you own the game in a way that you can't add it to Steam... sorry about your DRM, I guess?
The Steam Controller has always been described by Valve as an evolution of the Steam Deck tech, both hardwsre and software... it pretty much literally is a Steam Deck, without the screen and PC, in a different shape.
EDIT:
IRT to your edit:
Point 1: Correct. The Steam Controller... works through Steam.
Point 2: ... I don't think the first half of this is correct. I'm not 100% sure, but I think you can do this via Moonlight/Sunshine. I mean... I know you can in general stream Steam games from a PC using Moonlight/Sunshine, I'm not 100% sure this will work with a Steam Controller... but it works on a Steam Deck, so... probably will work.
As to the second half of this... yeah, the Steam Controller probably won't generally work as a controller for a game on a phone. Though FeX may actually somewhat/eventually aid with getting that to being possible.
Any projects? Make your project support Steam Input. I can't speak for other engines too well, but Godot has GodotSteam, supports Steam Controllers, the SteamInput system.
As to your internet related concerns: Steam has an offline mode. Unless you are running a game via Steam that has its own/extra DRM that requires a constant internet connection, you'll be fine.
SteamInput works without internet. If it didnt't, a SteamDeck's controls would not work at all without internet access. ... It does.
EDIT 2:
Just for super duper clarity here:
Making a Steam account costs nothing.
Maintaining a Steam account costs nothing.
Any ... .exe or .sh or whatever... can be added to Steam, to be launched and played via Steam.
The... only kind of situation where this wouldn't work is essentially via a game that is installed/managed by some other platform that basically encrypts the exe in a way that only that platform can decrypt.
A GOG game, or something from Itch.io, with no DRM?
Plop it in Steam, it'll work.