What you are describing has to do with a proposition being 'a priori knowable' rather than 'analytic'. There are many sloppy presentations of these concepts that tend to run them together, but they are distinct concepts and both are distinct from 'necessarily true'.
There are at least four different accounts of analyticity. The first is due to Kant, who coined the term. His idea is that with some propositions, the predicate is already contained within the subject. So, for example, "all bachelors are unmarried" might be said to be analytic because the subject 'bachelor' already contains the property 'unmarried'. The problem with this reliance on the concept of 'containment' is that it is too narrow: it doesn't cover cases of sentences that are not in simple subject-predicate form. Frege proposed instead that a proposition can be considered analytic if it can be derived from a logical truth by substitution of definitions. So, we can start with "all unmarried men are unmarried", which is a logical truth, substitute 'bachelor' for 'unmarried man' and arrive at "all bachelors are unmarried". This is much better, though it depends on the use of a particular (ideally formal) logic to make the concept precise. The logical positivists preferred instead to think of analyticity in terms of meanings, or linguistic conventions. On such an understanding, a proposition is analytic if it is true in virtue of its meaning, or in virtue of the linguistic conventions that govern the words from which it is formed. These accounts are broader, though even less precise, than Frege's.
A priority is concerned with whether a proposition is knowable independently of any empirical evidence, beyond that of merely understanding what the proposition means. Necessity is concerned with whether a proposition is not merely true, but could not be otherwise. Thus, analyticity is a linguistic concept, a priority is an epistemological concept, and necessity is a metaphysical concept.
It is important to keep these separate, because substantive philosophical theories depend on the relationships between them. The logical positivists claimed that all a priori knowable propositions are analytic, that all necessary propositions are analytic, and that analyticity provides the explanation of why some propositions are regarded as a priori or necessary or both. This is a non-trivial claim about the relationships between the concepts.
Kripke's theory of metaphysical necessity cuts across this account by holding that there are propositions that are necessary a posteriori, and also propositions that are contingent a priori, and that neither has anything to do with analyticity. Quine rejects the account on the basis that analytic/synthetic is not properly a well-defined binary distinction. Other philosophers such as Paul Boghossian hold that there is a relationship between analyticity and a priority, but not between analyticity and necessity.
All of which is a long preamble to saying that your proposed account might serve as an explanation of what it is for a proposition to be a priori knowable. If a person understands S, then that is all that is needed to know S to be true. On its own, it doesn't really advance our grasp of a priority much, unless perhaps you can embed it in a more substantial theory of what 'understanding' amounts to. And of course it is subject to all the usual objections as to whether there really is such a thing as a priori knowledge at all.
