Let's continue with you example of France DGAC.
National body: DGAC
DGAC is in charge of French ATC, in airports and enroute, manages ATC education and training (ENAC, École Nationale de l’Aviation Civile), is the spokesperson for operators, crews and industrials.
DGAC in general is the operational safety people you can see at the airport, EASA is not present. They are reporting to French authorities (e.g. Ministère de la transition écologique) who are still responsible for overseeing aviation activities in France.
Regarding safety, the domain of EASA:
DGAC is responsible for ensuring the safety and security of French air
transport. On one hand it is a regulatory authority, but it also
supervises security and safety [...]
DSAC, The Civil Aviation Safety Directorate is the department of DGAC
in charge of safety. Its task is to ensure safety for everyone
involved in air transport. It is in charge of monitoring the
application of the regulations, which are primarily developed by the
European Aviation Safety Agency (EASA).
STAC (Services Technique de l'Aviation Civile) is a branch of DGAC in charge of planning navaids and airports and designing airports systems.
EU body: EASA
The actual domain of EASA, as name implies, is safety. While EASA doesn't replace local authorities, it is the main regulatory body in EU for civil aviation.
DGAC (actually DSAC) on the other hand is participating to safety regulation preparation as do other national agencies in EU. Remember EASA was only 400 people a few years ago, maybe twice today.
EASA has issued a number of "regulations". Regulations don't have to be implemented locally contrary to "directives", they are binding per se. An analysis of the regulations structure is available here:
Some activities have logically shifted from DGAC to EASA, in particular the ICAO role to issue airworthiness directives for commercial aviation:
From 15th February 2006, DGAC France stops issuing Consignes de
Navigabilité (= Airworthiness Directives) on behalf of EASA
However ULMs or gliders topics (among other) are still under DGAC responsibility.
EASA vs FAA
Contrary to FAA which encompasses nearly every aspect of civil aviation in the US, EASA has a limited, though critical and expanding, role around security, safety and airworthiness.
Contrary to FAA, EASA is not an ANSP. Eurocontrol itself has not to be compared (yet) with FAA ANSP role, they actually cover only a very small part of the EU sky (Maastricht UAC), even if the long term idea is to standardize ANSP tools in EU and allow a true single sky ATM to cope with the increasing traffic. DGAC (DSNA branch) has still full authority for this aspect in France, and probably for some time since ATC means also coordinating with national air force.