Yes, I do it like that but my characters still mostly have preferences, skills they used before the story began and/or skills they really svck at. I use alternative life mod and one of my favourite starts is left for dead, and if my character used to be a good archer he ain't gonna track down a bow the moment the game starts. He'll just wander the wilderness, or look for a cave or a town or any other place to sleep at, and if warhammer is the only weapon he finds in the mean time it's going to be his main weapon for the time being. However as soon as he finds a weapon that suits him better he'll switch it out despite having the initial used weapon's skill being higher (for example, an archer who likes light weapons will use two handed as a last option where it'll get switched out for one handed with sword being the best option, and of course, the bow that he'll use almost exclusively the moment he finds one even if that skill is still at 15 and two handed is already at like 30).
I did have several characters who weren't any kinds of warriors before the story began, and those either stuck with a 1st weapon they became good with or a weapon that they felt like it suits them the best (without me having a pre-determined favourite).