What this gives the EU is the hammer to hit persistent abusers of user data. They want you to be careful with user data and not treat it like you own it; you do not. It is not yours to sell or share or publicize.
Edit; note as well that every country has a compliance office; if they know you are in complaince as in you are ‘good people’ (best effort, no giant holes etc; just best practice in our field which you should do anyway) they will not bother you with every (or any) user complaint after that. I have good experiences with this with far grave (and potentially criminally punishable) matters in a few EU countries.
The regulator is the effective judicial remedy.
In the UK there's also a First Tier Tribunal and probably an upper tribunal. These are when the regulator has made an error in law.
https://gdpr-info.eu/art-12-gdpr/
"The controller shall provide information on action taken on a request under Articles 15 to 22 to the data subject without undue delay and in any event within one month of receipt of the request. That period may be extended by two further months where necessary, taking into account the complexity and number of the requests."
Regulators. "Member State shall provide for one or more independent public authorities to be responsible for monitoring the application of this Regulation".
So, 28 countries, each with 1+ organizations. So you could find yourself having to deal with multiple parties in different languages.