Actually,Osteopaths are referred to properly as ‘physicians’, and they will adjust your spine (using entirely different techniques than a chiropractor - they seem to roundly despise chiros), as well as an entire range of treatment modalities, right up to and including surgical interventions.
That said, no, getting your bones all lined up won’t fix a hangnail…but it’s rather amazing sometimes what effects can be achieved, just by making a single muscle relax, or relieving a single impinged nerve. You could have a pain in your foot that gets cleared up when the vertebrae in your lower back are straightened out, so that your hips and legs aren’t trying to compensate structurally to the point where you end up with sore feet. Just because you may not consciously perceive a subtle misalignment or postural fault, doesn’t mean it isn’t happening or that it has no noticeable consequences - really, no different than failing to perceive a cancer you didn’t know you had, or any other disease.
I don’t want to oversimplify, or pimp for any one school of thought - but…knee-bone connected to the thigh-bone. It all works together, when you’re well, and not when you are unwell. Do you really care whether it’s a non-invasive technique or some drug that gets it done, as long AS it gets done without doing you more harm than good?