Of course, France today has provisions to prosecute someone in France for a crime committed elsewhere, which satisfies their European neighbours for the most part.
This said, a "bad" EU members might well choose to extradite further, once a prisoner is obtained from France. I expect this would trigger appeals at the ECJ and possibly repercussions for the "bad" country, but in the meantime the prisoner might well be gone. I also don't know what the "punishment" would be for a country that refused to execute a valid EAW. I guess this sort of thing is left to ECJ judges.
[1] https://ec.europa.eu/info/law/cross-border-cases/judicial-co...
[2] https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=CEL...
> unless they take over the execution of the prison sentence against the wanted person
Yes, that bit. France still cannot extradite its citizens, but it can meet its obligations to its partners by prosecuting and incarcerating someone within France.