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Schwarz reflection is an analytic continuation construction
The reflection theorem of Harmonic and holomorphic Schwarz reflection across the real axis is a direct analytic continuation statement in the present language. If a function is holomorphic on the upper half-disc, continuous on its closure, and real-valued on the diameter, the theorem constructs a holomorphic reflected function on the full disc. The original and reflected elements agree on the upper half-disc. That agreement is exactly the overlap relation of Function elements and direct analytic continuation.
So Schwarz reflection is not a competing construction beside analytic continuation. It is one of its cleanest geometric instances.
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Sources
- Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 6 §6.5 (standard reference, not scraped)