> I know I'm connecting to  port 587 since I specified it explicitly.  But I 
> don't understand the relevance to my problem.

Exim took the port number 465 from somewhere in its config.
465 is the TLS-on-connect port.
In some other part of config you specified 587, but Exim nevertheless
tries TLS-on-connect.
Search for "465", post relevant parts of your config.
Better entire config.

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