The curseforge page says that the key "M" opens the GUI. But I tested it and it doesn't do anything. I go into Minecraft's controls to see if I can configure the key, and there is no such option to configure the GUI.

How do I open the GUI?

This feature is a beta feature, available only on last beta version. Older versions doesn't have the same system. See changelog. Current README describes you old versions of the mod.

Any chance you could update the mod for 1.17.1 to have this feature?

No chance I will, sorry. This would take too much of my time, and I already don't have much time to put into modding. But if anyone wants to do the backport, I'll be happy review it

Does biome-specific music work in 1.17.1 as long as I make the resource pack myself? Or is that feature something that only comes with the GUI in later versions of the mod?

It comes in latest versions only. See changelog