So, he does not trust third-parties with his banking details,
but uses a third-party to send emails with a complete detail of his account to an email-server (which might or might not be run by a third party).
The logic is lost on me.
Furthermore, the e-mails are sent in clear-text and readable by any endpoint on the way transporting the e-mail to its destination. It's akin to sending your information back to yourself by post card rather than stored in a file with a third-party.
The alert emails do not contain “detail of [my] account,” only the last 4 digits of my account (for Chase). Even if someone were to intercept those emails, they couldn't do much with them.