I also find instances of this phenomenon very interesting. One example I can think of is in differential geometry where statements about manifolds are often easier proved (and in many cases, so far only proved) by routing through Riemannian geometry (since via partitions of unity, a manifold can always be given a Riemannian metric) and manipulating the manifold primarily through its metric structure (eg Ricci Flow) - even though the original statement doesn’t invoke metrics at all!
I also find instances of this phenomenon very interesting. One example I can think of is in differential geometry where statements about manifolds are often easier proved (and in many cases, so far only proved) by routing through Riemannian geometry (since via partitions of unity, a manifold can always be given a Riemannian metric) and manipulating the manifold primarily through its metric structure (eg Ricci Flow) - even though the original statement doesn’t invoke metrics at all!