Although the workaround I mentioned in the answer update did work, I was not happy about manually fetching those values. The solution for me was to remove the line
smtpClient.UseDefaultCredentials = true;
from the original code I posted. It turns out that smtpClient is initialized with the default credentials I set up in the web.config, and the above removed line was overwriting them with empty strings from CredentialCache.DefaultCredentials. I still don't know why CredentialCache.DefaultCredentials is empty or when this is supposed to be populated from the web.config, but this was the source of my problem.
If anyone has any further insight into this please post a better answer!
