I mostly shoot indoors, so what is the advantage of HSS?

You have full control of the ambient light if you, for example, shoot close to windows. With HSS you can control all ambient light and be in full control of the lighting. There is no need to use ND filters on the camera to control the ambient lighting. If you shoot some type of action in mixed light, it is great to use HSS to freeze the movement. Using a regular flash may frequently result in motion blur since the sync is normally not faster than 1/250s.