I actually found the answer to this question already, and just want to document my finding.
The problem has to do with using Node.js to access CouchDB through HTTPS. The CouchDB server has been configured with a SSL certificate generated by Let's Encrypt. The cert_file specified in local.ini contains both the server certificate and the issue's intermediate certificate. When I verified an URL through the browser, the connection was shown as valid. However, whenever I tried to fetch the same URL from Node, an UNABLE_TO_VERIFY_LEAF_SIGNATURE / unable to verify the first certificate error would be thrown.
I tried adding the root certificate to Node using the NODE_EXTRA_CA_CERTS environment variable. That failed. Then, I tried adding the intermediate certificate to NODE_EXTRA_CA_CERTS, and it worked. I could stop there, but somehow I knew that something was wrong.