No because the bolded bit isn't true.
The term profession enters French from Latin meaning one essentially referring swearing an oath, and originally referred to professions where you had to do that - doctors and lawyers, for example.
But over time, it came to mean "any job which required extensive training over years" and where you were independent (again, doctors, lawyers, etc. are theoretically independent), and also specifically not involved in a craft or industry (because those were regarded as something different and lesser).
Then English nicked the term.
So nothing that is not:
A) Independent
B) Not involved in a craft or industry
and
C) Requires extensive training
Could be a profession in the true sense any time before like, the 20th century. Also money changers as an actual job appear much later than forms of currency. So they wouldn't even be near the beginning even if we back-apply the 20th century English meaning of profession to antiquity.
The main point is, money has nothing to do with whether someone is professional or not. Absolutely nothing at all. People had jobs that required serious learning and hard work and even swearing of oaths long before money was really "a thing", and frankly people will still have professions in a barter-based post-apocalypse where money is dead again.