I don't know the details of IMAP well enough, but isn't the proxy what allows them to inject HTML into the email that iPhone's Mail app sees, but not any other client?
Yes - they need to perform what's called a Man in the Middle Attack (MiTM) to inject HTML into your email.
Normally, your iPhone (and other clients) retrieve email from Gmail's servers using the IMAP protocol. To inject content, Linkedin setup a security profile which placed themselves in the middle so you connect via IMAP to their servers, they fetch the content using IMAP from Google, inject their content, and feed it back to you.
This is why the email is not permanently changed. Only changed en route to your iPhone.
Normally, your iPhone (and other clients) retrieve email from Gmail's servers using the IMAP protocol. To inject content, Linkedin setup a security profile which placed themselves in the middle so you connect via IMAP to their servers, they fetch the content using IMAP from Google, inject their content, and feed it back to you.
This is why the email is not permanently changed. Only changed en route to your iPhone.