I have a similar problem on my house, built in 2011 and purchased inexpensively due to the leaks. My leaks are more front and center because they are all on a wall where the roof drains.
Disclaimer - I'm not a pro by any means. I've just spent my life buying houses that the git'er done flavor of contractors made a mess of and figuring out the correct way to fix their mistakes. Read a lot about stucco over various surfaces, and I think I now understand it pretty well, but a really good stucco guy is definitely going to know better than I do.
I would avoid trying to get a painter to fix a stucco problem

I would also avoid cementitious stucco on an ICF wall - even skim coats of it are heavy and you need a plan to hold up that weight. ICF's are great in many respects, but holding up a heavy wall treatment are not something they are designed to do as well as a framed wall with foam sheets. It may be possible to do a cementitious skim coat with a proper bonding agent application first, I don't know, but I kinda doubt it.
The question is then whether your stucco contractors are giving you good advice. How consistent is the advice between the 7 you've talked to? If they are all saying the same thing, then probably it's good. I would watch out for contractors who view one foam wall as the same as any other, what is under the foam is important because it changes your available options. If there's a big variety of methods suggested, then you need to use your own judgement.
Your synthetic stucco solution sounds about like what I plan to do on my house. Only difference is that I'm considering some expansion joints as well. One guy I talked to said they use joints on synthetic coat walls when it's more than 50' or so. The failures I see on my place are consistent with lack of a joint on the large runs, but also complicated by what are clearly adhesion problems.
I would also suggest going as smooth as possible with the finish, even with synthetic products. Really sandy, rough surfaces tend to hold water and soak it up. You want the rain to sheet right off the wall as quickly as possible.
The last thing I would say is to make sure you have drainage away from the wall. The problems could have started due to ponding at the base and caused continued failures from there. Good drainage never hurts (especially in AZ where summer rains can give you huge amounts of water very suddenly!) and the time to fix any drainage is while fixing the stucco or right afterwards.