I tried this and WP Mail SMTP. After configuring them the test (and live) emails and inspecting the raw message headers they all still show as originating on the web server, then going THROUGH the smtp/email server, and finally to me.
I assumed using SMTP directly would mean the web server (hosted on WP Engine if it matters) wouldn't show up in the header at all and it's look like it started on the email server. I definitely not and email/smtp expert (although I thought I knew how to read email headers).
Does this seem correct? My goal in all this is to use our authenticated SMTP server which has SPF records for it.