This is not an answer to your question.  I just wanted to point out that there are plenty of  examples of complete Einstein manifolds of infinite topological type.  I am aware of the following two papers at least:

1. [Complete Ricci-flat Kåhler manifolds of infinite topological type][2], by Anderson, Kronheimer and LeBrun

2. [Continued fractions and Einstein manifolds of infinite topological type][1], by Calderbank and Singer




  [1]: https://arxiv.org/abs/math/0508614v1
  [2]: https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-125/issue-4/Complete-Ricci-flat-K%C3%A5hler-manifolds-of-infinite-topological-type/cmp/1104179632.full