We have to be careful here about what we mean by "field." A field is a mathematical object that has been found to be very useful in making physical theories. I don't want to get bogged down in definitions, so I'll take a more conceptual approach.
An example could be basic quantum mechanics, where the state of a quantum system is represented by the abstract vector quantity called a "ket." <PERSON> PhD laid down this theory, which he used to show that <PERSON>'s wave mechanics and <PERSON>, <PERSON>, and <PERSON>'s matrix mechanics were two different algebraic representations of the same abstract object.
Now, a physicist or mathematician trained in field theory will say that ANY theory can be cast in terms of an effective field theory. And this seems to be true. But this is also true about writing any theory in a covariant form, even if the equations themselves are not fully invariant under general coordinate transformations (i.e. Newtonian gravity can be expressed in covariant form too, but only <PERSON>'s equations are fully invariant under general coord. trans.). So it comes down to what perspective do you want to have?
There's a rich history in modern quantum theory about whether it's better to use particles or fields. It traces its origins back to the classic debate between <PERSON>'s wave theory of light vs <PERSON>'s corpuscles of light. This article is a nice, short discussion that highlights the main points of contention for the modern debate between <PERSON> et al. and <PERSON>. about QED. 682
Together with <PERSON>, they shared the Nobel prize in 1965 for the synthesis of their ideas that <PERSON> arrived at independently.
However, do not be fooled by this synthesis! There are currently something like 9 equivalent formulations of the laws of quantum theory today, some of them do and some of them do not use the concept of "fields."
Along these lines, another example of a purely "non-field" theory could be <PERSON> and <PERSON>'s absorber/emitter formulation of electrodynamics. Even though they still talk about electric and magnetic "fields," these fields interact by emitting a kind of particle, rather than having to probe the fields using test particles like in classical electrodynamics. So they're unlike other usual "fields."
Lastly, one more example can be made with <PERSON>'s general theory of relativity. In its classic form, it is considered a "field theory," hence they are called "<PERSON>'s field equations." <PERSON> and <PERSON> successfully created an effective field theory version of <PERSON>'s equations, which has proven useful in quantum gravity research. So again, it comes down to what perspective is most useful for what you're trying to do.
A nice philosophical discussion in the modern context is here, where the author concludes that healthy, active areas of research should involve a mixture of field theoretic approaches with other approaches. I tend to agree with this attitude, since it never hurts to have several perspectives to understand a theory through - it actually almost always helps to advance the theory! Even if its just rewriting it in a different mathematical framework, which is essentially how <PERSON> discovered his relativistic equation for spin-1/2 fermions.... by rewriting the <PERSON> equation in a different way.
This is my 2 cents, but I bet others have good, and probably more technical, insights.
Your questions are good questions, but they are not well defined. For example, no one knows how to precisely differentiate between "effective field theories" and "fundamental theories." Indeed, we don't really even know what makes something "fundamental" anyway, this is something that is still up for debate.
Also, gravity as we understand it is still non-renormalizable, probably for good reasons though. So your first question doesn't seem to be the optimal question to ask. To me, it's more interesting to ask, "could we have explained the precession of Mercury's perihelion with an effective field theory instead of needing <PERSON>'s theory?" My answer is that it's unlikely to happen like that, basically because in practice the effective field theory is made after the more "fundamental" theory is already established. That is, by definition, an "effective theory" does not explicitly claim that the mechanism employed in the theory has a direct counterpart in the actual causes of the observed phenomena to which the theory is fitted.
EDIT: However, this has happened numerously in the past. For instance, before <PERSON> showed that quantum tunneling can explain how the Coulomb barrier is overcome in stellar nucleosynthesis, stellar models assumed that nuclear processes produced the right amount of energy that was observed without explaining those processes. 800