For sure, but even then the EU needs to go through the democratic process it has in place. If you try to introduce a law guaranteeing press freedom it'll be vetoed by Hungary.
The question then becomes how much you can achieve with the laws on the books. Traditionally cases will be brought before the Court of Justice, which will then have to decide if it can or should interpret existing laws in an expansive manner in order to cover whatever issue is in front of them. Judicial independence in Poland is a good recent example.
While the Court of Justice could introduce democratic safeguards this way, there's also a certain irony to having democratic safeguards imposed by a supranational organ of 27 judges nobody knows with minimal democratic legitimacy. If the Court of Justice oversteps there's also no guarantee the affected member states will accept its authority, and enforcement mechanisms are limited.
The Court of Justice has generally been pretty successful of pushing an integrationist agenda, but it's only so much it can (and should) do. And if the political organs are deadlocked, the EU is basically a lame duck by design.