I don't take offense, I just suspect that I'm using "generalist" in a different way than you.
That said the limitation may be what you're watching rather than anime itself. The shows I listed all have experienced heroes from the start and I can list many more.
This Link leads to a list of 27 of the top anime from the 90s. Of those I think maybe three or so have inexperienced protagonists at the start and the rest are the kind of rugged, battle-hardened experienced types I think you want.
That said even some pulps have newb heroes just starting their journeys. Star Wars, for instance, is quite Pulpy in feel.