If you go down the path of using a third party mailing list provider, I've been fairly happy with MailChimp (aka
MailKimp), though some recent policy changes there made me also consider MailerLite. MailChimp seems to have grown into a big corporate behemoth focused more on retargeting marketing, sadly.
If you switch to a third party, you'll also need to look into doing all the SPF / DKIM stuff in your DNS that "authorizes" those services to send mail on behalf of your domain.