Whilst you have found a plugin to handle notifications separately, it’s not necessarily the job of POST SMTP to determine where and what emails are sent.
The main job of POST SMTP is to provide a different mail transport other than the in-built PHP Mailer function. i.e. it’s there to just send emails.
POST SMTP does not care who created the email, it just sends whatever it is told to send.
For example:
WordPress Core decides who gets notifications about the core (updates, comments etc).
A contact form should have the necessary options to allow you to route notifications to the form submitter and to a site administrator.
A shopping cart should have the necessary options to allow to route sales notifications to the purchaser and to various other admins.
POST SMTP does not need to know about what settings the Core uses, nor does it need to know anything about Contact forms, shopping carts etc. It simply takes what ever would have been sent to the php_mailer function and re-routes it to whatever mail method you have decided to use.
So in your set-up with WooCommerce, it is the WooCommerce that should format the sales based emails and determine when and what emails are sent. You can set up different ‘funnels’ based on certain actions within WooCommerce. POST SMTP doesn’t have a clue about what WooCommerce is doing other than it receives a mail to send and sends it to addresses that were supplied by WooCommerce.
If sales emails are going to website-admin@yoursite.com instead of going to sales-team@yoursite.com, that is down to WooCommerce not being set up correctly, rather than POST SMTP (or any other SMTP type plugin)
You don’t need a separate plugin to add the behaviour you want, you need to correctly configure WooCommerce.
POST SMTP does have the ability to override certain fields and prevent plugins from changing them, but in most site set-ups, it will be the job of WP Core and any plugins or filters defined by the site that determines what and where emails are sent.
In a perfect website, you should think along these lines. You install POST SMTP, set it up to send emails and then forget it was ever there. That’s how it should work.
The advantage of POST SMTP however is that it has extra features such as a fall-back method in case your primary sending method fails. The ability to not only your own SMTP servers, but dedicated mail providers (mailgun, gSuite GMAIL etc). It also provides the ability to log all outgoing emails to allow you to debug problems and even re-send a failed message (or one that the user somehow deleted or their own mail provider didn’t deliver).
