Intuition has limits--it's of no use when we're faced with situations dissimilar to what we have faced before. Logic has different limits--it only works on things that are well-defined, because otherwise you get caught in meaningless semantic arguments. The strength of one approach covers for the weakness of the other, if we can direct both at a single problem.
Axiomatizing systems is an approach for taking what we know from intuition, and using logic to extend that knowledge to cover things we don't have experience for yet. There are other approaches, but they're fuzzier.
Axiomatic systems have two major weaknesses: They're fragile, and they can be unwieldy. By fragile, I mean that classical logic doesn't deal with consistency well. Like, at all. Like, a single inconsistency, no matter how small,
renders the entire system invalid. By unwieldy, I mean that systems with large numbers of axioms are difficult to reason about, because they have more stuff in them.
Both of these problems can be alleviated by reducing the number of axioms you're dealing with. The more that you can reduce "intuitive axioms" to "theorems" by proving them from other axioms, the simpler your system becomes. This makes it less likely to be inconsistent, and easier to work with because there's less facts to keep straight.
Simpler systems also make it easier to test things that you don't know if they're true or not. You can try to test if a fact is true by seeing if it interacts with other axioms in a way that creates a contradiction. In simpler systems, it's much easier to pinpoint which actual a contradiction arose from, and then you can evaluate which of the two axioms you want to discard.
Even in an example as synthetic as the real number line, this gets important when e.g. you want to start talking about infinities. Infinity is inherently a non-intuitive concept, so any attempt to intuit how it ought to behave is going to lead to contradictions. If your foundation is a giant pile of "intuitive facts" about how numbers behave, it's really hard to add infinity to the mix and keep them all straight. You'd have to gut and replace like half your damn system in order to work in infinity, and even then you won't be sure you've done a complete job. When your system is built on like 5 rules, it's easy to change one rule to admit infinity, and then study the consequences in a logical fashion.