The "Back doors" in AXE are a slightly different thing. Many countries have laws that says that law enforcement have the right to wiretap phone calls under some circumstances. This means that telcos want and ask for this feature so that they can comply with the law. The telcos are aware of the systems capability because they need it to be there. Anyone who reads the law can see that the telcos has to facilitate wiretapping, but they obviously dont want to advertise it, so its an open secret.

Its quite different if you deliver a solution, with a hidden back door that the customer dont know about or have asked for, for the benefit of the intelligence service in the country of manufacturing. Enabling a nation to wiretap illegally in countries where they do not have jurisdiction.

How does Linus ability to go public make it more dangerous for intelligence services?

That implies intelligence services approaching him with such an offer would put themselves in danger of consequences just for asking, when that's evidently not true: There were no consequences for the NSA for asking Linus, there were no consequences for the NSA in the vast majority of cases it was caught doing something it shouldn't do.

It's also not true that in the "free society" is some binary thing: In the US National Security Letters are a very real thing, outfits affected by them are legally not allowed to publicize it. By any definition of this "free society keeps intelligence services in check" logic that should not be a thing, but it is, just like many other legal tools to suppress the free and open reporting about "national security" issues.