Maybe you just have misinterpreted what staging is reffered to in different contexts:
In context of letsencrypt staging certs:
As far as I know he LetsEncrypt Staging Authority issues exactly those kind of certificates that you mentioned. They are not trusted by browsers, but only used for initially testing if issuing certificates works in general. After that works you need to switch to letsencrypt production authority.
In context of your staging API:
It does not mean that for your staging environment you use the letsencrypt staging authority. Instead you also have to switch this to the production authority.
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Maybe you just have misinterpreted what staging is reffered to in different contexts:
In context of letsencrypt staging certs:
As far as I know he LetsEncrypt Staging Authority issues exactly those kind of certificates that you mentioned. They are not trusted by browsers, but only used for initially testing if issuing certificates works in general. After that works you need to switch to letsencrypt production authority.
In context of your staging API:
It does not mean that for your staging environment you use the letsencrypt staging authority. Instead you also have to switch this to the production authority.