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, by Anderson, Kronheimer and LeBrun

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

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, by Anderson, Kronheimer and LeBrun

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