Issue resolved, I’m leaving here some more info in case that’s usefull to someone else.
I’m not sure what resolved the issue unfortunately. I tried the configuration yet another time today, and now it works. I didn’t change a thing to what I’ve done before. Home Assistant went through a few updates in the meantime, perhaps one of them helped.
Before that unexpected success, I was about to retry but this time I planned to reboot HA and clear my browser cache because the issue I had (on a service, click on start, loading animation appear on the button and then the service appears to be stopped, despite the logs showing overwise) has been encountered by other users and that was the solution they found.