1. If the theorems of ZFC are the only statements that you consider intuitive, then proving a statement independent of ZFC would require adopting an axiom that is not intuitive.
2. If there are statements independent of ZFC that you consider intuitive, then those could be adopted as axioms to form a stronger system.
3. If you want a system that can prove unintuitive statements that are independent of ZFC, then you may or may not be able to form such a system by adopting an 'intuitive' axiom, depending on what you consider to be intuitive.

But I imagine you are asking about something broader. It can be difficult to ask questions on such abstract subjects if you are not familiar with them, but if you are able to be more specific, or if you have any clarifying questions, I will try to help.
I'd say mostly yes, statements independent of ZFC will be unintuitive, for selection reasons.  ZFC is the set of axioms most mathematicians work with because they and the conclusions they lead to are what most people consider to be intuitive and reasonable without much controversy (though choice is more controversial than most of the others).  People took all of the mathematical things they considered to be intuitively true and found the smallest set of axioms they could make that would contain them all.  If there were other intuitive axioms that were independent of ZFC, they would have already been included as well.

This is not a universally true law, because different people consider different things to be intuitive.  What you consider intuitive will differ from someone else.  But will tend to be true and is not a coincidence.
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A lot of work by set theorist is about introducing intuitive axioms. So it's not about introducing any new axioms. There are a few general idea about what is considered intuitive, one of them being the maximality principle "a set exists if there are no logical contradictions to it existing". So it's not necessary to introduce unintuitive axioms, the new axiom could be intuitive depends on your opinion of whether an axiom of intuitive or not. However you definitely need new axioms.

There are plenty of statements that feel so intuitive true that are independence. Like "if a proof of P exist then P is true". For example, Grothendieck do things that are not allowed by ZFC, because it's much more convenient that way and what he did feel entirely harmless, ie. essentially introducing new axioms that are intuitively true.

Even ZFC axioms themselves are not free of controversy, not everyone consider them intuitive. Axiom of Choice is infamous. But power set is controversial as well. Different people have different ideas of what is intuitively true.
I wonder if talking about a metaphor for independence would help.

If we started with two "axioms", X and Y, and we visualize them as vectors in the plane (and specifically we're focusing on the the integral vectors, like graph paper), the unit vector along the positive X axis and the unit vector along the positive Y axis, and we derive all possible "conclusions" from X and Y by summing them, and summing their sums and so on, then we get X + X, X + Y, Y + Y, and so on. A whole lattice of "conclusion" vectors comes into existence in the upper right hand plane. This operation is taking the "closure" of the axioms with respect to the operation of summing.

Some vectors, like X + Y, are not in the original axiom set, but they are in the closure, and so adding them to the axiom set doesn't change the closure. They're consistent, or admissible.

Some vectors, like -(X + Y), are both not in the original axiom set, and also, if you add them to the axiom set, then suddenly the closure expands to be a particular, familiar, boring, "everything", pattern. These are called "inconsistent".

Notice that X + Y and -(X +Y) are opposites. Furthermore, often, opposite pairs have this pattern - one is admissible, the other is inconsistent.

However, some vectors, like X - Y or Z, when you add them to the axiom set, make the closure somewhat bigger (they're inadmissible) but they don't make the closure expand to the familiar, boring, "everything", pattern (they're consistent). Also, their opposites (Y - X or -Z), are inadmissible and consistent.

These vectors are analogous to "independent" statements. Generally any time that you expand the language (talking about Z when we were only ever talking about X and Y before) then you will get independence.

If you are willing to expand your axiom set and also its closure, and you're trying to make sure a particular statement (which is not in ends up in the closure, you can always add that particular statement - then you have a one-line proof of that statement; it's an axiom. Is there another way to prove it? Well, if you imagine taking out a piece of graph paper and coloring the vectors, there's a sort of "cone" of admissibles that goes up and to the right, and another "cone" of inconsistents that goes down and to the left, and there just aren't that many remaining spots with no color near to the origin. X - Y and Y - X are the only two. You CAN break down a proof of 2X - 2Y into multiple steps using a new axiom, X - Y, but you CANNOT break down the one-step proof of X - Y in the same way. There isn't any room remaining.

So in the more complicated logical situation, I imagine high-dimensional "cones" emerging from (subsets of) the  ZFC axioms, and their opposites, and they are staining a vast number of short statements either admissible or inconsistent. If you go outward far enough, towards the high-dimensional "waist", you start getting to some unstained "independent" statements.

My guess is that some of the currently discussed independent statements are also pretty close to the origin, that is, some of the shortest ones. Making a map of what this "waist" of ZFC overall looks like might yield results like "according to this metric of statement length / complexity, there are no shorter / simpler statements that could be added to ZFC to demonstrate (for example) Martin's axiom. That is, the shortest / simplest statement you can add to ZFC that gets you to a system where Martin's axiom is true is Martin's axiom itself."

Of course, what I've done here is replaced "intuitive" with "short" or "simple", which might not be what you want - regardless of whether you think that's a legitimate move, it sounds like an interesting research project!

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