I really have very limited experience of this, so please forgive me if I appear to be a spokesman from the Ministry of the Bleedin' Obvious.
I haven't found that ingrained dirt is a problem with guttering, fascia and soffits. The dirt is very loosely attached and just needs a fairly light brushing and a light water flow to simply loosen it and rinse it away.
The problem that I have had (and I've only done a couple of these jobs) is how to get 3" long bristles into a 4" deep crevice to clean the top of the fascia behind the gutter and the tops of those brackets. The brackets seem to get covered in spiders' webs which trap the dirt and look even worse when everything around them is gleaming.
My only solution is to get up close and personal with a microfibre and give it a hand job - so to speak.
That's why I've only done bungalows and had to turn down a few offers of work from regular 2-storey houses.
It gives me no pleasure right now to be turning down work !!!!