The first rule of flying with medication is that it goes in the carry-on, over everything else. If you can't fit so much as a book alongside your meds? Tough, meds have priority. In the US at least, the airline simply isn't allowed to make you check your medication, and the TSA is not allowed to confiscate prescription medication in its original container. OTC stuff, beyond what you need on the actual flight, should just be purchased at the destination.

FedEx in that context is probably because it has less trouble getting through customs, rather than for airline employees.