Mainframes (Z/OS) will mainly run batches (most often COBOL programs controlled by JCL jobs) but there is also CICS (https://en.wikipedia.org/wiki/CICS) which runs COBOL (and probably others) "on demand".

My experience with connecting UNIX to Z/OS was not with a real API (HTTPS) but UNIX > TIBCO EMS > something called TIBCO Substation > CICS. It was horrible, mainly because of Substation which needs to translate on the fly between two completely different platforms.