Probably the black dots are where the tougher hairs were broken off.

What you've got to watch is that hair and skin are very similar in composition and what's designed to destroy hair will damage skin. And what's strong enough to destroy coarse hair growth will almost certainly damage the relatively sensitive skin of the face.

That applies to electrolysis and laser treatment too, as well as to all chemical depilatories. They all depend on striking a delicate balance between destroying hair and not causing serious damage to the skin and all work best where the hair is fine and slow-growing and not hormonally-stimulated.

Plucking and waxing distort the root and eventually cause the hair to grow at an odd angle and be much more obvious.

My moustache hair is coarse and so for me the best answer is shaving.