Concerning your example, there is definitely no analytic proof of the existence of rational curves on Fano manifolds. This is one of the dream of complex geometers... You can also consider this weaker statement: given a Fano manifold $X$, can you construct an entire curve (i.e. a non constant holomorphic image of the complex plane) in it by analytic methods? Even this is not known...
On the other hand, no algebraic proof is known for invariance of plurigenera for varieties not of general type (the analytic proof is due to Siu, later refined by Paun, and the algebraic proof for varieties of general type is due to Kawamata).
