DefaultCredentials Doesn't Turn Your Windows Login into a Basic Password
Match .NET's default security context to the proxy's authentication scheme instead of expecting a browser's stored Basic account to appear automatically.
The browser remembers a proxy account. Your .NET client uses CredentialCache.DefaultCredentials and receives 407 without the retry you expected. Those two facts are compatible.
DefaultCredentials represents the current security context for supported integrated authentication. Microsoft documents its applicability to NTLM, Negotiate and Kerberos. It is not a request to extract a username and plaintext password from the browser's saved logins.
If the proxy offers Basic and requires a separate account, configure that account through the proxy credential interface the client supports. An explicit NetworkCredential and the current Windows security context are different inputs.
Check the advertised challenge before changing the password. Also check which user context runs the application. A desktop executable and a service can have different identities even when launched on the same computer.
There is no reason to copy the account into the destination's credential property as well. Keep the intermediary's login with the intermediary. The useful diagnosis names three things: the selected proxy, its requested scheme, and the kind of credential the application actually supplied.