Thank you @pgr!
I have implemented the fix, and later updated to 7.11.10, but the API still gives me 404.
Further searching in the forum brought me to the conclusion that the issue could be related to the fact that I use nginx and not Apache, meaning that the htaccess file is not interpreted correctly.
Now my challenge is to find a working nginx configuration for SuiteCRM; I will try first with this suggestion: suiteCRM nginx config
Cheers,
Guido