Davidson Obsession

Davidson proposes (in "Truth and Meaning") that for a language to be learnable by finite creatures like us there must be a finite collection of axioms which entails all true statements of the form '"Snow is white" is true if and only if snow is white'. Then he, and his followers, argue that people with various kinds of theories can't satisfy this constraint e.g. that nominalists can't get a theory that entails the right truth conditions for mathematical statements without using axioms that quantify over abstracta.

Something about this argument strikes me as fishy, and I've spent hours obsessing over it at various times, replacing one putative "refutation" with another. :( But I can't stop thinking about it, so here's my newest attempt.

First, grant that for someone to count as understanding some words they need to know all the relevant instances of Tarski's T schema. So they have to be disposed to assent to every such sentence. Now, as every sophomore seeing the Davidson for the first time points out, it's trivially easy to make a finite program that 'assents' to every query that's instance of the T schema in a given language, or enumerates all such instances. But Davidson requires more, there needs to be a finite collection of axioms, which logically entail all the instances. This is what gives Davidson's claim its potential bite. But now, we ask, why think this?

EITHER
Davidson thinks that to know the T schema you need to be able to consciously deduce them other things you antecedently know. In this case the requirement that each instance of the T schema must be deducible from a finite collection of axioms would be motivated.

But this can't be right because no one can consciously produce such an axiomatization for our language. If we learned the T schema by consciously deriving it from some axioms, we should be able to state the axioms. Therefore, conscious deduction does not happen, and cannot be required.

OR
Davidson allows that it suffices for each instance of the T schema to individually feel obvious to you, (and for you to be able to draw all the right logical consequences from it etc.)

But to explain the fact that each sentence of this form feels obvious when you contemplate it, we just need to imagine your brain is running the sophomore-objection program which checks every queried string for being an instance of the T schema and then causes you to find a queried sentence obvious if it is an instance. Once we are talking about subpersonal processes there is no reason to model them as making derivations in first order logic, so the requirement is unmotivated.

Perhaps Davidson might argue that the subpersonal processes doing the recognition are somehow doing something equivalent to quantifying over abstracta, so the nominalist, at least, would have a problem. But do subpersonal processes really count as quantifying over anything? And if they do, is there any reason we have to agree with their opinions about ontology?