Rationality justifies rationality recursively: Rationality works because, by the rules of Rationality, it works. So, in a sense, yes, everyone has to choose their axioms.
The first missing piece of the picture is The Lens That Sees Its Flaws ( https://www.lesswrong.com/post... ). A system can be self-improving and continue to approach accuracy, even if it starts in an imperfect state.
The second missing piece is that if you pick any sort of axiom like "believe in things that have been shown to work in the past", you get Rationality. You can of course be obtuse: anti-inference has been wrong every time before, so by the rules of anti-inference it is bound to be the correct system! But if you make that claim, you're rejecting the common ground of "believe in things that work". Communication only really works within groups that can at least approximately agree on certain axioms, and most humans have agreed with that one.
> Someone says there are more oranges than grapes.
"more" is a quantity word, not a weight word. In standard English conversation, this means that "count(oranges) > count(grapes)". One person is right, and one person is wrong. If you want to discuss a different quantity, you can do that, but you have to specify: you're allowed to say "there's a greater mass of oranges than grapes".
> You are insisting that counting (reason) is the only legitimate way to arrive at the truth.
No one is insisting that one is more "valid" than the other - merely that the conventions of English are used differently depending on whether you care about count or mass. There isn't even a disagreement here! If you say "but the mass of the oranges is greater", any rational person would agree with you, and you should be able to agree back with them: "yes, the mass is greater, but the quantity is less". These two observations aren't contradictory, just measuring different things.
> It is however the one you have faith in.
If you want to communicate, you have to play the same game everyone else is playing.
If you don't believe in rationality and reason, why go on a forum and try making reasonable arguments? Wouldn't "squid purple smiley-emoji" be just as convincing?