I do treat this idea, albeit somewhat briefly, in §5.2 of my SEP entry. A better source is §3 of Richard Zach's SEP entry on Hilbert's programme, since it presents the evolution in Hilbert's thought more comprehensively, with reference to specific texts from Hilbert's corpus.

Hilbert's programme changed in the later 1920s, from one emphasising consistency to one emphasising conservativity, in analogy with the distinction in physics between the theoretical and empirical parts of a physical theory. Giaquinto (1983) is an important early paper on this shift in Hilbert's thinking.

Maybe have a look at Hakli and Negri's paper 'Does the deduction theorem fail for modal logic?', doi.org/10.1007/s11229-011-9905-9

You write that "you don't just know that a theory is consistent, just because its axioms are true" . Even assuming a fairly weak notion of truth (namely, satisfiability in some structure), the soundness theorem guarantees that the truth (i.e. satisfiability) of a theory implies its consistency, at least assuming that the proofs in question are formalisable in some fixed deductive system for which the soundness theorem holds.