You have run into one of the main themes of contemporary logic: the difference between "truth in the standard model" and "provability". This is an extremely deep issue, so I'm sure other people will also have something to say about it.

The difficulty with focusing only on standard models rather than on theories in general is that essentially the only way to convince someone else that the standard model has some property is to prove it, and then you're back to the problem of choosing axioms. For example, the reason you know π is transcendental is because you recognize certain axioms that are true in the standard model and which allow you to prove π is transcendental. If someone else did not already believe π is transcendental, you would try to convince them by getting them to accept the axioms you used to prove it.

In some cases, we can make a set of axioms that completely describes a standard model. For example, there are complete axiomatizations of Euclidean geometry, which allow you to prove any statement in the language of geometry that is true about the standard Euclidean plane model and disprove any statement false on it.

But for other models, like the standard model of the natural numbers, there are theorems that show we can never find an effective, complete, consistent axiomatization. The axiom systems we use to study these models are called "essentially incomplete". For these models, it isn't clear in what sense you could make the underlying concept (e.g. "natural number") precise enough to eliminate independence results.

That being said, one nice property of models in classical logic is that any sentence in the language of the model is either true in the model or false in the model. So there is no concept of independence from a model. The downside is that for models like the standard natural numbers, it takes stronger and stronger axiom systems to determine more and more true statements about the model.