Error mail sent

Which error email are you referring to?
A post SMTP error message? Or a non-Post SMTP site generated error message?

If your account is receiving error messages of a certain type, but you want those to go to the site owner, could you not set up a filter on your mail server/mail account that would automatically forward those emails to go to the owner instead.

Error messages about a site usually go to the site admin of which there is usually one.
If you want to filter off certain messages to different accounts, using a filter at the admin account level is probably the quickest and easiest method to achieve that.

I’m assuming that as site admin, there are some messages that you still want to receive?

You could change the email in the WordPress->Settings->General tab, as that is listed as the site admin address. But that means ALL emails sent to that address by WordPress will go to that address, not just the email failures.

I’m assuming (again) that you only want email failures to go to the site owner and emails such as plugin/core updates required emails to go to you.

You have two options around that, on your own email account, set up a filter so that emails containing “Post SMTP email error” in the subject are forwarded to the site owner. (Bit clunky)

The better option is to go to the Notification tab on Post SMTP settings and change it from using “Email” as the “Notification Service” to “Pushover” or “Slack”.
That will require your website owner to set up a Pushover/Slack account so they receive notifications of email failures. The best feature of using this is that notifications are instant.
You could also have your website owner set up “Push to Chrome”. You’ll need the owner to install the Chrome extension. I’m not 100% certain if notifications are shown if Chrome isn’t running (but in most cases, if Chrome is installed, it will have a process running somewhere !).

At the moment, Post SMTP sends email notifications to the standard WordPress Admin email address as defined by WordPress’s own settings.
You could request a feature that allows a different email address to be specified to be used as a Delivery Notification Email Address, but it’s not available at the present.