It really is analogous to most govt fines e.g. speeding tickets: the government (the police) gives you a ticket, and if you pay it then OK, no court involved, but if you challenge it then the courts get involved.

But more generally, the government should and does get priority access to the courts already. Criminal courts exist solely to serve the government; you cannot bring criminal suits as a citizen yourself, only civil suites. Also, e.g. in Germany the government and legislatures (federal and state) get priority access to e.g. the constitutional (supreme) court. A mere mortal cannot just file suit directly in the constitutional court, but has to go through the lower instances first (unless there is something similar to a class action petition, showing a sizable chunk of the population sees the same issue and wants it decided). Members of the parliaments and IIRC of the cabinet are allowed to file suit in the constitutional court directly. The reasoning here is that if it was allowed for citizens to petition the highest court directly, then the court would do nothing else than write rejection letters for bullshit petitions. While the govt and legislatures incl the parliamentary opposition of course represent the people (in theory) and aren't stupid morons wasting the courts time (in theory).

PS: Google was already fined €50M for GDPR violations, and there is probably more of those in their future. Facebook got fined €10M so far, IIRC, also with more to come. And don't forget the billions of Euros worth of antitrust fines against Google and Microsoft, e.g.