I think that Jef is on the right track here…
“Reputation” has been mentioned several times in this thread. When a project is concerned about their reputation, trademark is probably the context in which that concern should be addressed. Trademark exists specifically to address concerns about reputation.
Especially for the purposes of this discussion, trademark does not need to be registered. If a developer publishes a trademark use policy for an unregistered trademark, I would expect that policy to be respected. Trademark does not need to be something that requires money and time, beyond the time required to publish guidelines for the use of the trademark.
Mozilla’s trademarks are registered, but I think they still serve as a useful example: Mozilla Trademark Guidelines
In part, they serve as a useful example because Debian has demonstrated how distributions should handle trademarks when the conditions for use of the mark are incompatible with the requirements of a downstream user of the source code. Debian’s requirements, for a long time, included the ability to patch vulnerabilities in the package they shipped, independent of the upstream releases. Mozilla’s trademark policy does not allow the use of their marks on modified versions of the software, so Debian simply shipped a build with a different name and logo. That is the correct way to use Free Software in compliance with a trademark policy that does not permit use of the marks to identify a modified product. Publishing the build with a different name and logo protected Mozilla’s reputation, and communicated to the community that the build that Debian provided was Debian’s build, not Mozilla’s build.
This is another case of what I said earlier, that one of the things that’s valuable about Fedora is that it demonstrates how software development and distribution work. Debian also provides those demonstrations.
I don’t think it makes sense to publish source code and call it Open Source or Free Software, at the same time you tell people not to build and distribute it. But it does make sense to say (via a trademark policy) that the software can be used but not your name. That’s reasonable and common.
I am … mostly certain that is not true of trademarks. You can publish works that contain trademarks under a license that allows distribution, while simultaneously asking that your marks not be used for identification, via a trademark policy.
P.S.:
Asking Fedora not to publish Flatpaks would mean that Fedora cannot serve as an example of the right way to build and publish software, and I think that would be very bad… that’s one of the most useful things that Fedora does.