Hi David, thank you you for the quick answer,
I wasn’t aware of this feature and it looks like a solid explanation for my problem, thank you !
I was probably misleaded by the fact that, apart for the background image, other changes I was doing to the header applied even when the static page name was set as display condition (but I probably mixed those with the classic Generatepress cusotmizations I also made).
I just made a test by switching my front page then trying again the display condition to the static page and it displayed. So my issue seems pretty much solved, thank you for the help !
best regards from France