Naturally, I solved the problem five minutes after posting.

The problem was unrelated to exim, per se.

I just switched from courier. couriertcpd was still running in the
background and attempting to pass control of the connection to /usr/
sbin/couriersmtpd, which doesn't exist any more.

What remains to be explained is why netstat reported that exim was
listening on port 25 if it was actually couriertcpd listening on it.