"We are worried."
on the same line of thought .... the majority of sales are made by focusing on a worry (a problem) the potential buyer has (even if you have to 'makeup' the problem or exaggerate it by framing it in front of their eyes) .... a sale is often made based on the level of "worry" the product or service will resolve and often not based on what the actual thing being sold is.
Solicitors are not directly from the "worried" meaning.
Solicitors (from the French Solliciteurs), from the verb Solicit, from the French Solliciter, meaning to ask for something with insistance, to solicit.
It has no link with the meaning "worry", or not wanting to talk.
Sollicitors/solliciteurs are people who ask something, maybe with insistance, because it's important to them.
The meaning, and its evolution, is well described in this page:
Old meaning for "solliciteurs": "one who conducts matters on behalf of another". It was because people was send to ask on behalf of someone else, that they were called solicitors.
You can't skip studying the old French/French meaning with English etymology. English is (almost) never directly from Latin, words had the time to get other additional meanings.
Sollicitude also exist in French, with a meaning closer from the Latin meaning. Meaning to worry for someone, with a lot of compassion.
Solicitous comes from the old form Solliciteux, that is not much used now, but still exist, full of "sollicitude".
"Worried" is only one way to translate sollicitus. It can also be translated "moved," "agitated" or "concerned."
In English, "solicitous" means to be concerned or attentive. The use of "solicitor" for an attorney is British English, and I'm not familiar with the history of that usage. In the U.S., as you say, a "solicitor" would be someone concerned or attentive about gaining your business.
The etymology is not 100% correct. The meaning in English and Latin are a bit far, as the old French changed the meaning.
A solicitor is absolutely not from Latin (at least directly), but from French.
When you say "a solicitor would be someone concerned or attentive about gaining your business", it's an interpretation, it could be right if the word was directly from Latin.
Soliciteur/Solicitor was someone who made the action of "soliciter".
It comes directly from this verb. Of course, it's from Latin, but the interesting bit is the meaning in this old French term (not Modern French solliciter, with 2 "l" and another meaning), the meaning of this verb "soliciter", as it's the direct root.
Soliciter: Meaning "one who conducts matters on behalf of another"
So U.S solicitors are people who conducts matter on behalf of someone else, at least, etymologically. It's a very old word.
The dictionaries explain that: "A solicitor may also instruct barristers and represent clients in some courts."
The meaning is here, in "represent".
The old French lose the Latin meaning, and passed this changed meaning in English.
It's the same thing when you consider an Indo-European root, it can mean something, the meaning can be changed in Latin, and we inherit of the changed Latin meaning, not the genuine one of the PIE root.
For "sollicitus" meaning "agitated", according to the dictionaries, it's the first meaning of this word, and it applies to innert things, like the sea.
The second meaning is worried, applied to people. It's the figurative use of the image of the agitated sea.
There are no evidences that the first meaning, non figurative but literal, could be applied to people.