Texture Space properties on Bazier Curves don't do anything? Trying to UV map hair texture on curves

Welcome.

See the documentation.

“Texture Space” affects the generated coordinates, not the UVs.

No. UVs on beveled curves are always (0,0) to (1,1) - there is no way to adjust this on the curves panel - you can only do it through Geometry Nodes or with a mapping node in the Shader Editor (or Python).

For personal projects you are expected to bake everything to export to a game engine. Consider something like Bake Wrangler with a Convert-To-Mesh Python script or with GN if you want to automate it.

Good luck.