Lots of good answers here already, but I can also add my own perspective. Fundamentally, you're right that there isn't really anything "special" about radicals. The reason I personally find the unsolvability of the quintic by radicals interesting is that you can solve quadratics, cubics, and quatrics (that is, polynomial equations of degree 2, 3, and 4) by radicals.
To say the same thing another way: the quadratic formula that you learned in high school has been known in some form for millennia, and in particular you can reduce the question of solving quadratics to the question of finding square roots. So (provided you find solving polynomial equations to be an interesting question) it's fairly natural to ask whether there's an analogous formula for cubics. And it turns out there is! You need both cube roots and square roots, and the formula is longer and uglier, but that's probably not surprising.
Whether you think n'th roots are "intrinsically" more interesting than general polynomial equations or not, this is still a pretty striking pattern, and one might naturally be curious about whether it continues for higher degrees. And I don't know anyone whose first guess would have been "yes, but only one more time, and then for degree 5 and higher it's suddenly impossible"!
To say the same thing another way: the quadratic formula that you learned in high school has been known in some form for millennia, and in particular you can reduce the question of solving quadratics to the question of finding square roots. So (provided you find solving polynomial equations to be an interesting question) it's fairly natural to ask whether there's an analogous formula for cubics. And it turns out there is! You need both cube roots and square roots, and the formula is longer and uglier, but that's probably not surprising.
Whether you think n'th roots are "intrinsically" more interesting than general polynomial equations or not, this is still a pretty striking pattern, and one might naturally be curious about whether it continues for higher degrees. And I don't know anyone whose first guess would have been "yes, but only one more time, and then for degree 5 and higher it's suddenly impossible"!