I'm not sure I understand what this changes to the overall situation. Remember that a formal system also can't prove its own consistency, so you can't just ignore bits that you don't like and expect that it doesn't matter, it's possible that one of these "true but unprovable" propositions would actually render your system inconsistent, in which case it ought to be addressed.
I mean in the end Godel's own proof effectively relies on this: he builds a proposition using the rule of the system saying "this proposition has no proof" and at this point you either have to accept that the proposition is incorrect, but then your system is inconsistent (it can be used to derive falsehoods) or that it is correct but then the system is incomplete. So you may decide not to call it a truth but you also can't really call it a falsehood.
I would also wager that mathematicians generally prefer incomplete to inconsistent systems, if which case you have to accept that these unprovable propositions are indeed truths.
It's like the "does the set of all sets not containing themselves contain itself" paradox, you can't just ignore it as irrelevant because it puts into question your entire formal system.