Answer

First, if you give up adopting axioms of this kind, you won’t be able to know anything. Every claim is proven on the basis of axioms, and without axioms we have no information at all. So we have no option of giving them up. Are you suggesting thinking without axioms? You understand that there is no such thinking at all. What, then, would you rely on? On what you see? That too is a rash axiom—that our sight is reliable. Moreover, it really is not correct, since there are also optical illusions.
Our intuition is an uncertain tool, but we cannot do without it. Therefore we should relate to it with a measure of respect and suspicion. What my intuition tells me is true in my eyes, but I will try to check it and examine whether that is indeed so.
Euclidean geometry is evident from our point of view, and it is indeed correct for Euclidean space. The mistake lies in assuming that our world is Euclidean. That is an incorrect intuition. But it is important to understand that the deviation from Euclideanity is completely negligible. That is, this intuition too is also correct, aside from tiny deviations that in any case cannot be detected. It is exactly like the axiom that our sight is reliable, which I mentioned above. There are exceptions, but they are at the margins. Overall, this intuition is correct.