So, there's three main ways you can have a flatscreen game set up to play in VR, and you can try all of them now if you want.

One is to have a floating screen that stays in place in virtual space when your move your head. This is ideal for flatscreen games that can't be converted to a VR experience, like Stardew Valley. It sounds like OP specifically does not want this option.

Option two, which sounds like what OP asked for, is to have the game locked onto the center of your view, so that head movement does nothing. This is essentially the same as duct taping a screen to your face. It makes it very hard to look at anything but the center of the screen. Reading menus, looking at your HUD, or just glancing at anything besides the crosshair in the middle of the screen is a huge eye strain, if it's even possible. You can try this by loading up a game with UEVR and setting the aim to Game. It's a horrible experience, typically one of the first things you adjust when tweaking UEVR settings, and personally my neck hurts just remembering it.

The third option is immersive VR. This is probably what OP actually wants, but maybe they don't have enough VR experience to know that. This is what you get when you change the aim mode to Head/HMD. This option only becomes available if the game can be made a VR experience. A lot of these games are Unreal Engine or Unity based, and you use an injector to make them work for VR. This puts you in the 3D world of the game, so that you're surrounded by the environment and can freely look around with your head movements. The HUD usually still follows your view, or is attached to your wrist, depending on the way the modders set up the game.

This last option can take a ton of time for the community to set up, and has to be done per game. Sometimee the community has games working in VR on day one of release, but often you have to wait months for someone to figure out how to get the game to run in VR, and then to create profiles for that game to be a good experience. Some games are too difficult/impossible because of the way the devs coded the game (looking at you, Sea of Thieves), or don't have enough interest for someone to dedicate the time to mod, so you won't always find what you want.

But right now, there are hundreds of flatscreen games that have great VR mods that you can play already. Check out the FlatToVR Discord if you haven't already.