Well, this got pretty long, because there's lot's of important distinctions that we usually don't draw attention to.
To summarize, there are (at least 3) kinds of unproven statements.
The first, is as you say, statements that resist refutation. Both their truth value and undecideability value are unknown to us. Providing a refutation resolves both questions, "it is a decidable statement and it is not-true".
Continuing to not have a refutation has more possibilities than just "we haven't proven it true or false yet".
The statement could then be proven to be undecideable: which means we prove it neither follows from nor does it contradict our existing axioms in the system under consideration. It is here that we are safe to extend the system by just adding it as an axiom.
We might do so for fun, or we might do so because we have reasons, outside the system, to believe the statement really is true. Incompleteness demonstrates just this; we can't prove it ... but we really do think it's true!
Incompleteness informs us that we can never have a set of rules that prove "all true facts" about numbers, even if we keep finding undecideable statements and each time have reasons to believe it's true and so allowing more true proofs about numbers (to extend the "true facts" we know). But only because we are finite.
If you're talking about the platonic world with all true statements, mathematical or just statements in general, then incompleteness isn't a problem.
The pattern already emerges from the existing axioms without explicitly adding the pattern as an axiom. So, what do you gain by adding it? — alcontali
I'm not sure I would phrase things the way you do (as undecidable statements are not necessarily existing patterns), but adding a "true unprovable" statement as an axiom allows the system to be extended by then proving more things with the new axiom.
For instance, the axiom of choice is not provable from the other ZF axioms (it neither contradicts ZF nor is implied by ZF nor is it a pattern that emerges from statements following from ZF), so we can add it and extend the system. This is probably the most famous example, but we can also just take other axioms away from ZF and have the same situation and then add them back in to extend the system, to demonstrate the process (taking an axiom out, doesn't make the things depending on that axiom suddenly false, they become suddenly undecidable); something controversial like the axiom of choice (the controversy being not that we can extend ZF but whether we need to for the practical problems of engineers) isn't required to see this "adding axiom" process happening, that's the only way to get a set of axioms: staring with one axiom and then adding the next.
This "extending the axioms" also occurs whenever problems with actual values are worked out with defined relations or values. Saying X really is 5 extends our axiomatic system. This may seem trivial and irrelevant, but I'll get back to it at the end (we may have reason to really believe X is 5; i.e. that X represents the number of people in the room, and the number of people is really 5: it is a true statement we are adding to the system, as an axiom, that we could not prove with our previous axioms; if we are using this statement in our formal system it is no different than the other axioms, it is only us, outside the system, that knows it is a different kind than the others and stated to be true for different reasons; so we don't call them axioms, but they are formally they are the same thing and we demarcate that with "if"; i.e. "if x = to 5" then we are going to treat that as an axiomatic statement ... for now).
Therefore, I really do not see what you would gain by adding the Riemann hypothesis to the axioms of number theory. — alcontali
The reason we don't axiomatize the Riemann hypothesis is because, unlike the above two examples, we really don't know if there are no counter examples; if we add the hypothesis as an axiom and a counter example is found then now we have a contradiction and all statements (all statements!) and their negations can be proved; we want to avoid that. When the Riemann hypothesis is "assumed" it's being used as an axiom same as every other, but again, we outside the system know it's not like the others, and if a problem arises (a contradiction appears) we won't be pointing the finger at axioms at random, we'll know who needs to go.
For the Riemann hypothesis to be added like the axiom of choice, it would need to be proven that it's undecidable; i.e. independent of ZFC. Then we could add and make ZFCR or it's negation or do neither no problems, and it's a practical question whether it's useful to add it and it's a philosophical debate whether it is really "true" (formally speaking, in the same way we can ask if ZFC axioms are "really true"; though from outside the system we can understand ZF is different than the C which is different than R) -- useful to note here most mathematicians no longer debate whether axioms are true or not; you do what you want and you see what happens, if you want to do something useful pick useful axioms ... of course, that's not how mathematics is taught. (Another good example is imaginary numbers, we need to add i squared equals 1 as an axiom -- and for that matter analytic continuation from which the Riemann hypothesis arises is also itself an extension, adding more axioms because we feel like it; why it's such a focus is that a bunch of other stuff about prime numbers and number theory become true making a deep and unexpected connection; this is what's irksome, having to jump through all these hoops to be really, really close to proving things we have not the slightest clue how to do otherwise).
Anyways, once we understand all that we can extend our mathematical axioms by making new axioms from undecideable statements, and we can imagine axiomatizing all knowledge through this process as the OP suggests. For instance, "at what time you'll wake up tomorrow" is not provable from our current set of axioms, but once it happens we can add it as an axiom to a giant formal system we're continuously extending as we think of new statement we prove are undecideable but think are true anyways as well as experience new things that get dropped into the system and true because they happened (they didn't have to -- i.e. we couldn't prove it from previous axioms -- but it did happen and so becomes a true statement we can use as an axiom). A "perfectly rational" being with "all the axioms" would indeed see all the conclusions in the axioms and experience knowledge in this way (an omniscient being would have no subtleties about what's true and false; a perfectly rational but not-omniscient would just have perfectly accurate probabilities that follow from any uncertainty in their axioms; i.e. we can interpret "all the axioms" in an omniscient way or in a way of perfectly setting up all the experience the being has as axioms and making perfect inferences).
Of course, we're far from being able to do anything remotely close to this.
This is why I describe Kant, not only because of the historical parallel, but because the fact we are so far away from experiencing "real knowledge" in the way the OP suggests (that I agree, "real perfect knowledge" works like that; all the conclusions are understood simultaneous to the axioms) "our actual experience" of knowledge is the moral effort required to understand a tiny, small, miniscule part of the "platonic" world of all truths. Because we can get it wrong along the way gives rise to moral tension. The Kantian philosophy is that there is a path -- there are true axioms that can be discovered and we can through effort conform our behaviour to those axioms approaching, in steps no matter how small, the world view of perfectly rational beings for whom it is just obvious and there is no tension -- which is opposed to nihilism of no true axioms existing (at least morally), skepticism of not true axioms being knowable but they maybe there, relativism of one form or another where true axioms depend on oneself (in a circuitous and unresolvable way ... unless it is already resolved), divine consequentialism (true axioms are decided by God and only true by being told to do it or suffer the consequences to disobey), utilitarianism, scientism, emotivism (where things aren't resolvable at all).