TPM, your son is very observant. It is very tough to do it for an extended period of time. Some players can do it but most struggle with the role of DH. Let me share some numbers with you. My son became a regular starter on the mound when conference play started last year. When he was not pitching he was the DH. He never caught ANY SEC games. His batting average during the SEC games was .227 with 4HR and 18RBI.
This year he was a catcher and NEVER DH’ed. During the same SEC period this year his batting average was .387 with 8HR and 30RBI.
Like TPM says it’s tough to set on the bench and think about your hitting…. plus everyone else has a safe harbor by running out to play defense and the lonely DH has to listen to the coach explaining what you did wrong during your last at bat.