Texture paint on a non-tiling non-UV mapped image?

Why am I not allowed to paint in the viewport with this node setup?

Almost worse, with this setup a stroke is not displayed in the viewport until you release the mouse/pen!

Is there a different node setup I can use that doesn’t rely on UVs (hard requirement) which allows for texture painting in the viewport?

No. Texture painting in 3D View is based on UVmapping.
There are many different ways to UVunwrap a mesh.
You can paint a texture that is not UVmapped in Image Editor.
If you want to paint an image in viewport that is not a shading texture but a modifier texture, UVs are also required.

Only other way to paint something in viewport is to paint attributes instead of textures.
But in that case, a special mapping would be ignored, too.

That’s unfortunate, but thank you for the information!