This is actually a great set of questions, part of which you have already answered for yourself!
As you have surmised, a molecule is a molecule, no matter how it is made. Acetylsalicyclic acid (more commonly known as aspirin) has the exact same effect on your body, regardless of whether is it isolated from willow bark or synthesized from salicylate. People who worry about the source of medicines may say that there is some by-product or containment left over after synthesis, but they don't realize just how much purification is required to legally sell a drug in the US.
From a marketing standpoint, a substance can be called "natural" if it is derived from a natural source. So, vanillin that is extracted from vanilla beans can be called a "natural flavor," no matter how much processing you put the vanillin through. But vanillin can also be made from compounds found in clove oil, spruce trees, or petroleum. Vanillin molecules made this way would have be labeled "artificial flavor." (C&EN had a great article on this: The problem with vanilla | September 12, 2016 Issue - Vol. 94 Issue 36 | Chemical & Engineering News)
Compounds made from "bad" sources are no more safe or dangerous than the same compounds made from "good" sources. Articles suggesting otherwise are misinformed. You can make an argument for sustainability concerns--most plants are a renewable resource, whereas petroleum is not--but that does not tell you anything about the safety of the product itself.
I hope this helps.
