1. Self-host on a reputable VPS provider. Deliverability is usually not a problem if your provider actually takes action against spammers on their network. They tend to work pretty hard to keep up the reputation of their IP space. I have self-hosted my own email for well over a decade on providers like these with no serious issues. It's not generally too hard to test the reputation of an IP/domain before putting it into production.

2. You can use a dedicated SMTP relay service, but these are usually quite expensive and their customers tend to be bulk email senders for blasting out marketing wank. You probably don't want your domain to be associated with these anyway.

3. Another option is to buy a full-fledged email account from any of the common providers and just use their SMTP servers. (Make sure to set your SPF records appropriately.) Not expensive but typically far from free. And it seems silly to self-host your mail if you're literally paying a company to provide the same service...

4. This is less "self-hosted" but the cheapest and most reliable way to get your email out is to sign up for a cloud account at any of the major providers and just use their SMTP relays. Most will allow you to send from outside their network, after proper authentication is set up. Unless your monthly email volume is north of 4 figures, it will likely be free or cost pennies per month.

Don't forget that amongst email providers reputation is a thing, and if you go out and purchase a domain, it may be some time before you can actually use it. Lots of providers specifically penalize domains younger than X months old, they will "graylist" IPs that haven't talked to them before, and a few will flat-out blacklist entire gTLDs known to be heavily used by spammers.