What made scientists start to think about what was previously 'forces-exerted-by' (Newtonian view) to fields (e.g. electric fields and magnetic fields)?
The concept of field lines originated with Michael Faraday, who didn't have sufficient mathematical background to understand how brilliant it was.
I have only skimmed this article, but it appears well researched and seems to address your question: Origins of the Field Concept in Physics, by Ernan McMullin PDF. My reason for nominating Faraday is that I was addressing the concept of a field as a "form of matter" as Hans Ohanian puts it. The concept of field in fluid dynamics is more mathematical than physical.
Scientists needed a theory to explain why forces could be conducted through a vacuum. There is nothing inside the vacuum (i.e. no mass) and therefore only a field concept can explain why an electrostatic or magnetic force or even gravitational force and be transmitted through "nothing".