Toolset Form notifications are just sent using the built-in WordPress wp_mail function, it doesn't have any kind of bespoke function to send emails.
I've been looking into this all morning, testing notifications with the plugin (which I confirm don't work).
This is the line in our plugin code responsible for sending the message:
$isSend = wp_mail($to, $subject, $body, $header);
(wp_mail is a core WP function: https://developer.wordpress.org/reference/functions/wp_mail/)
I don't know when or how the WP Mail SMTP plugin intervenes in the process, but that wp_mail call returns true (i.e. the mail is dispatched successfully). What happens after it is is out of our hands.
I even ran Xdebug to step through the code and step into the core wp_mail function to follow what happens, and it ultimately arrives at this line:
$send = $phpmailer->send();
...which also returns true (i.e. reports that the mail is dispatched successfully).
I can't account for what WP Mail SMTP does that breaks this.
But I did try an alternative SMTP plugin, Post SMTP (https://wordpress.org/plugins/post-smtp/), and I'm pleased to report that the notifications are sent fine in that case.
So I suggest you switch your SMTP plugin to use Post SMTP. (If you want to continue with WP Mail SMTP you may want to contact their support for help, being sure to mention that Toolset Form notifications work successfully with Post SMTP.)
Also, I recommend updating the notification settings of your form to use the same From address as your SMTP provider (in my case I was using Gmail).