What purpose do the rubber 'hairs' have on new tyres? Up close you can see at least three 'hairs' of rubber from this picture from the side of my tyre:



Why do they exist? I've heard they are just a bi-product of the manufacturing process.
Note the technical term for what you are calling "hairs" is "vent spews."[1] As the name indicates, they are a consequence of an important detail of the tire fabrication process.

While liquid rubber is being injected or otherwise forced into the tire mold, air bubbles can form in the rubber or between the rubber and the mold. These bubbles can cause the finished tire to be weakened or malformed. To prevent that, tire manufacturers place vent holes in the mold to allow the air to escape under pressure. Necessarily, some rubber will get forced into these holes, forming the vent spews upon curing (which is the process by which the tire rubber converts from liquid to solid).

These vent spews serve no useful purpose in the finished tire; conversely, they can be removed without consequence.

[1] for some citations, see this ask.com post and this reddit post