Thinking about sentence structure in Japanese. Given that the simplest sentence in Japanese is a stand-alone verb, would it be valid to say that the rest of the sentence is an adverbial clause, and that the particles simply define how the rest of the sentence modifies the verb?
Well, depends on what you mean by "adverbial clause". If you try to interpret that term extremely formally in a linguistic sense, no, it's not.
But, I think you're generally on to something:
Predicates (which can be verbs, i-adjectives, or nouns/na-adjectives + the copula), take:
arguments (parts of the clause that are necessary to understand the predicate), e.g.,
"John will go."
adjuncts (parts of the clause that aren't necessary to understand the predicate), e.g.,
"_ will go quickly."
Both types of things can fundamentally be understood as "modifying" the predicate, which I think is what you were trying to get across with the usage of the word "adverb".
However, there are other types of modifications possible which don't really fit into that box. For example, relative clauses such as
"the store [I went to yesterday]"
are not modifying a predicate (since there is no predicate), they're modifying a noun phrase (in this case 店/"store").
There are also many other constructions, for example conditionals,
"If I go to school, I can see my friends."
where it's much more of a stretch to understand them as directly modifying the predicate (会える).
Perhaps you should consider taking a linguistics course if this sort of thing interests you? :-)