I think you are conflating two different uses of the word "true".
First, in any formal system there can be sentences that are true in some models and not in others. It simply makes no sense to ask whether those sentences are true in some purely abstract sense. By analogy, the statement "The house is painted green" is true of some houses and not true of others. There's no reason to think we should be able to assign a single truth value to such a statement, and no reason to want to.
Second, it is sometimes (but not always) the case that we have a particular model in mind from the outset. So if we've all agreed in advance that the house we're talking about is the one located at 1600 Pennsylvania Avenue in Washington, DC, then it does make sense to ask about the truth value of a sentence like "The house is painted green", and there is no ambiguity about whether that sentence is true or false.
In particular, when we about arithmetic, we usually (but not always) have one particular model in mind, namely the standard model of the natural numbers (i.e. the counting numbers that you learned about in kindergarten). Therefore, statements in, say, Peano arithmetic, are unambiguously true or false (as long as we all realize that we're focused on that one model). That notion of truth coincides with Tarski's more formal definition. We determine whether such statements are true or not by studying the natural numbers. There is no single "usual" formal system in play here, so I don't know what you mean by the "usual means" for discovering which sentences are true. Sometimes we use Peano arithmetic. Sometimes we use induction up to epsilon-nought. Sometimes we use Grothendieck universes.
Therefore, in response to your questions: When we talk about arithmetic, there is no "usual means" for determining what's true and therefore no need to develop "unusual means". Moreover, it would certainly be way counterproductive to adopt as system of multiple truth values when every sentence we can state is already either unambiguously true or unambiguously false.