So I just solved my own "SMTP connection failure" error and I wanted to post the solution just in case it helps anyone else.
I used the EXACT code given in the PHPMailer example gmail.phps file. It worked simply while I was using MAMP and then I got the SMTP connection error once I moved it on to my personal server.
All of the Stack Overflow answers I read, and all of the troubleshooting documentation from PHPMailer said that it wasn't an issue with PHPMailer. That it was a settings issue on the server side. I tried different ports (587, 465, 25), I tried 'SSL' and 'TLS' encryption. I checked that openssl was enabled in my php.ini file. I checked that there wasn't a firewall issue. Everything checked out, and still nothing.
The solution was that I had to remove this line:
$mail->isSMTP();
Now it all works. I don't know why, but it works. The rest of my code is copied and pasted from the PHPMailer example file.