I took a look in the software and it is not clear when this might have changed for you. According to the changelog this went into effect in late 2016 so it is odd that you have not had reports from users until now! See https://github.com/wpsharks/s2member/issues/870
Here is the explanation (from the code - so it is probably in the S2Member help):
Custom Passwords are easier for users. However, this has an impact on site security. Enabling Custom Passwords disables the New User Notification email that contains a password-setup link for each user. In other words, enabling Custom Passwords effectively disables any sort of email verification procedure, because the user is allowed to set their password during registration, instead of doing that via email confirmation—which is the default WordPress behavior
So…in a nutshell…if you let users submit their own password on registration then there is no need to send them a verification email. As far as I can tell, the S2Member “Welcome” email is actually just a content customization of the native WordPress registration verification email. Don’t need to verify the user…no “Welcome” email.
@clavaque may have something to add. He is the new owner of the S2Member plugin.
Regarding not wanting to add another plugin…I have used this notifiation plugin in my standard plugin suite for over 2 years. No issues at present and only had one glitch in the past (coincidentally missing new user emails) which has since been fixed.