The results on open Riemann surfaces are not "rare". They are just well forgotten.
I only list a few books which deal with open Riemann surfaces:
MR0114911, MR0228671, MR0159935, MR0264064, MR1973182.
It is true that there are "too many" Riemann surfaces, and not too much can be said about
"all of them". However, there is a highly non-trivial classification, and some subclasses
are very important. For example, the class of "hyperelliptic" surfaces of infinite
genus plays a very important role in the study of Schrodinger operators and their
finite difference analog, see for example the works of Sodin and Yuditskii, MR1288838,
and Kean, Moerbeke, MR0397076. Abelian covers of compact surfaces is another class
of open Riemann surfaces that was studied a lot: MR0740581.

Added: here is a recent non-trivial result about arbitrary open Riemann surfaces: https://arxiv.org/abs/2103.16702.