They're probably talking about True Scotsman's Self-Hosting whereby your MTA is not configured to go through someone else's SMTP forwarding host, but is slugging it out on its own, sending mails directly to target mail domains, and is sitting on an IP address that doesn't have reputation for sending mail. Self-hosting like it's 1992.
The wise man accepts that using a reputable forwarding host for sending is still valid self-hosting.
I use Mailgun, but all the big name providers should be pretty much the same for low volume sending. Unfortunately my 9 year old VPS IP was already on a spam list when I got it, not sure if it falls off eventually.
Nope! I use my local Big ISP, and have for 13 years.
To quell your worries about a prospective service, you can try them without rushing to relying on it. Just point a mail client at their SMTP host and test for a while.
I imagine if you have your DNS records set up strongly enough it would warm up to your MTA's IP address rather quickly, but I have not tested this for a few years now.
The wise man accepts that using a reputable forwarding host for sending is still valid self-hosting.