Post History

You are trying to absorb too many centuries in the stride at once.

I don't know what happened between Latin and Middle French, but by the time the (French noun) "soliciteur" got derived from the (French verb) "soliciter", the meanings involved seems to be like "to ask/request with urgency", "to ask/request with earnestness", or perhaps "to ask/request with the proper status to have the request considered".  In short, a "soliciteur" was a professional pleader, someone capable of acting as someone's agent.  That's the meaning of "manager of (somebody's) affairs"; and as a special meaning, if the affairs were of legal nature, then it means the kind of a lawyer who can represent their clients at a court.  Middle English borrowed both the verb and the noun and some of those general and specialized (legal) meanings.

I suggest that you look up the etymology of "serious" from Latin "serius".  
Yes it means something unrelated, it's just a parallel semantic development for comparison.

The development from Latin "serius" (something important, potentially menacing) to modern English "serious" (something just real enough as to be difficult to completely dismiss) is perhaps a shift in an parallel direction to the development from Latin "sollicitus" to French "soliciter" - but the meaning travelled a shorter distance, and it is perhaps easier to pin point what the shift has been also because the mediaeval legal specialization of the term is absent in case of "serious".