It just means there is a framework for granting a request.
There are two types list treaties and dual crime. Dual crime means it has to be crime in both countries, list treaty mean applies to only specific lists of crimes
There are other restrictions that could be limiting, some countries won't extradite if the death penality is a possibility, some will not extradite its citizen's but will do visitors or residents etc.
As with all deals, there will be always some other items in play while the treaty was negotiated and signed, recognizing or supporting the other country's claim, trade agreements, tariff reduction etc. Depending on the size of the partner the deal will be favourable or not.
[1] For example Assange is Australian in UK being extradited to the U.S.
Very rarely, it does become political.
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.
The USA just happens to be more aggressive about seeking extradition than most.
The USA has taken an expansionist view to the definition of jurisdiction, particularly for network-related activity. In short, they basically claim jurisdiction everywhere. That's something that very few other countries can realistically try to pull off without people laughing in their face, and its success is entirely due to might.