the origin of the -man suffix
There are many words in English where the reference to “man” is from a Latin root, not a German[ic] one. In those words it means “hand”, as in “manual” → something done with the hands.
Perhaps words such as “chairman” are preserving the original Germanic meaning of “man”, viz “a human being (irrespective of sex or age)”, per definition I. 1. of man sb.¹ in the OED. However, many compound nouns are given as examples of the II. 4. definition of “adult male person” in subsection p., such as “barman”, “postman”, “salesman”. The latter definition of “man” was expressed in Old English with the word wer (which has survived to our day by hiding in “werewolf”).