Once again, this is a complex question. It does very much depend on the context and the attitude of the government.
To give you an example that I personally saw in Senegal, which is now a lower middle-income country, GPE has worked there with the government, and they have specifically in their education planning determined—I think they call them “darah” schools rather than madrasah schools, but it's the same concept—that they will reach out to those schools and seek to regularize them in the system. If they are regularized in the system, they will qualify for some government funding.
In that context, my sense is that those schools were not so monied and resourced by outside interests that the government funding would be not attractive to them. The schools did want the government funding, so they were prepared to do the registration and compliance work that would get the funding. The lesson from that, I think, is that many of these questions are questions of political will, government regulation, and system structure, and they are the very things that, through the GPE approach, we work on.
In context, while some of these schools might be very well resourced, I do agree with you there is a competition-style problem, but at the end of the day, governments can regulate and structure systems, including systems for non-government schools.