I think it's helpful to always remember that "governance" is primarily adversarial. It's only necessary when some individuals and sub-groups don't actually want to do the things the governing organization is demanding. As such, the motives are completely broken for a governing body to control the majority of press outlets.
The problem is, the motives are broken for ANY entity powerful enough to have a significant voice to control or provide primary information sources. As soon as it's useful as journalism, it's even more useful as propaganda and manipulation. This is driven by consumers, with producers following rather than leading the trend - no matter how cynical audiences become, they don't actually make better decisions on how media adjusts their beliefs.
That's my summary of 21st-century information dissemination: the goodheart cycle increased faster than the consumers could improve their epistemic hygiene, and there's no way to fix it (short of making smarter humans, aka raising the sanity waterline, which is an even bigger unsolved problem).
As one of their questions Reporters without Borders asks for their World Press Freedom Index among others:
Part of the case of the EU against Hungary is that its press is largely government-controlled or controlled by supporters of the government. Voices critical of the government have a lower share of the public attention. Philanthropically funded journalism that intends to provide critical media gets attacked as being funded by Soros and intended to manipulate the Hungarian people.
COVID-19 showed that there are similar dynamics in the United States and other European states where voices that are critical of the regime have a hard time being published . The fighting critical content for being Russian disinformation and Hungarian strategy of fighting critical voices for being influenced by Soros follows similar dynamics where outside influence is overblown and the narrative allows for acting against critical voices.
While some national governments have state media, the EU currently doesn't have its own media outlet. Given the EU perspective of the problems in Hungary, funding critical journalism would be a good intervention. If the EU would start its own media, there's the question of media governance. How could EU-funded public media be governed so that it will represent voices from the full pluralism of opinions of people?