Men's hair grows faster not for evolutionary direct-reason but evolutionary accidental carry-over. Testosterone is a growth-enhancing hormone of nails, muscle and even bone. These are indeed to do with protection and tenacity pre-fights and post-fights.
Nails are made from keratin, so is hair. That is the only reasom why males grow hair faster than females do.
Before you ask, hairier men in terms of thicker hair especially on areas that aren't necessarily helpful (ears, stomach, anus etc) are usually high in testosterone vs the norm but the correlation isn't direct. The genes regarding more pores of hair unleashing during and after puberty aren't the same genes linked to how fast a man's hair grows. There is indeed correlation because of coincidence since men with slower hair growth with more pore openings rarely benefitted from it (it also is taxing, men with very hairy bodies need more protein and calcium than men who have less, yet the slower growth wouldn't benefit them much at all).
You may assume being hairier was most useful in colder climates especially during the ice age but the correlation is just not there. Brown asians are the hairiest with blacks being second hairiest and neither thrived in cold climates. Inuits are some of the least hairiest of us all and obviously had cold climate as a significant factor during their evolution. It is simply related to sensitivity to testosterone.
Brown Asians are the race with the highest amount of females who are also hairy (we don't notice in this day and age thanks to shaving, waxing, hair-removal creams and lazer hair-removal technology). So their sensitivity to testosterone in the body even applies to females who do have a small amount in their bodies.