Is research a viable business model?
In Europe, companies exist that primarily live from FP7 or, now, H2020 funding. They rarely contribute much to projects in terms of research, but often handle tangential issues, such as project management, dissemination, marketing, or association with standardization bodies. Their model works because (1) they do things that none of the research partners actually wants to do, and (2) because they are often SMEs, which is generally politically helpful to get your funding proposals accepted. Further, as writing grant proposals is their core business, they usually also have grant writing experts that have the time to actually keep up with the various changes that the european commission is constantly applying to its programmes and rules (something that many researchers struggle to find the time for).

Formally, these companies often camouflage as either consultancies or technology spinoffs. That is, you will not find it on their web page that their core business model is actually to milk grant programmes, as this is strictly against official funding rules (as a company, you can only apply for funding if you plan to commercialise the results, and you are not supposed to make a profit from the project itself). In practice, the project officers seem to tolerate these businesses - for now.


  Can such a company be started (nearly) from scratch?


You will need a reasonable amount of seed money, as the H2020 rules require proof that an organization applying for grants has enough financial stability to likely survive over the typically 3-year project duration. I know from one startup that wanted to participate in a project (for actual technology reasons) which was in fact struggling to cross this hurdle. However, if you have the required seed funding, I see no reason why you would not be able to start such a business. Whether it is a smart business idea is a different question, given that you are basically living off of a loophole in the current funding practices, and you can never tell how long this loophole will stay open.

Edit:

I just saw that I missed this question:


  Would it be feasible to request funds to continue the development of this PoC (still very far from a commercial product) as a start-up or that simply doesn't make sense?


Yes, at least in Europe there are funding sources for exactly this kind of case. You should inform yourself about the concept of Technology Readiness Levels (TRLs, slide 4), and the different kinds of funding available for each TRL. If you look at this presentation and go to slide 32, there is an overview over different funding instruments in H2020 depending on the maturity of the idea.