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I don't find the definition of a conductor clunky.

Consider any fibration f: X \to Y, just between the algebraic varieties for now. Let's say we want to study it somehow in algebraic geometry. One of the first things that will come to our mind is the locus of points L on Y where the fiber degenerates. It's an important data, and we will be able to work with f as smooth fibration outside of Y\L, e.g. we can then consider representation of \pi_1(Y\L) on cohomology of a fiber.

Now back to elliptic curves. They are in fact schemes over Spec Z (bear with me if you don't know all the words, it just means they are like X and something else is like Y in the above example) and the denegeracy locus is the submanifold of Spec Z given by equation N = 0, N being the conductor. Away from 2 and 3, this N is defined by this property and being the smallest number.

So I think from a geometric point of view , it's clear that the conductor for elliptic curve was bound to appear somehow. Now there are, of course, strange and mysterious things about it, that why it is especially how to relate this definition to the same as in other definitionsone involving modular curves, but that's kind of next thingstep.

The same procedure actually applies to number fields, since the conductor, again, is the ramification locus on Spec Z of the map of Spec R there (R being the field's integers). However, as was pointed out by some people, the conductor of a field associated to CM elliptic curve is not the same as the conductor of elliptic curve itself!
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I don't find the definition of a conductor clunky.

Consider any fibration f: X \to Y, just between the algebraic varieties for now. Let's say we want to study it somehow in algebraic geometry. One of the first things that will come to our mind is the locus of points L on Y where the fiber degenerates. It's an important data, and we will be able to work with f as smooth fibration outside of Y\L, e.g. we can then consider representation of \pi_1(Y\L) on cohomology of a fiber.

Now back to elliptic curves. They are in fact schemes over Spec Z (bear with me if you don't know all the words, it just means they are like X and something else is like Y in the above example) and the denegeracy locus is the submanifold of Spec Z given by equation N = 0, N being the conductor. Away from 2 and 3, this N is defined by this property and being the smallest number.

So I think from a geometric point of view, the conductor for elliptic curve was bound to appear somehow. Now there are, of course, strange and mysterious things about it, that why it is the same as in other definitions, but that's kind of next thing.