I send a couple newsletters (one tech related, other for marketing for a coffee shop/bakery I own), and it was fairly easy to land on inboxes. Now, I don't know with a 100% certainty that they were delivered, but if DMARC records and bounces are anything to go by, self hosted emails aren't all that hard.
When you buy a VPS, it helps to check if the IP you receive is already blacklisted. After that, it's easy to use some outside service to monitor DMARC responses and blacklist. There are about 40 black list databases. My domains and the IP address are on none of them.
Realistically an absolute fuckload of spam and not a lot of genuine email comes out of residential and cheap VPS hosts IPs so they already have a negative rep.
The good reputation IP range is the biggest problem. I have a more expensive ISP at home that takes its network seriously, so I look more like a commercial interest in that regard. There are decent services that will act as relay for you and probably cost less than such an ISP (or a similarly more expensive VPS provider that takes priority care of related things) if outgoing SMTP is your main reason for considering one.