There are any number of email sending providers with API access - Sendgrid, Mailgun, Mailjet, etc. Amazon SES is the most cost effective, but you'd need to do more setup youself vs. one of the API senders.
I'm with this person. Let the professionals do the heavy lifting in terms of email delivery. You just worry about setting things up correctly in terms of DKIM, SPF records etc. The services mentioned all have dashboards to see the amount of spam complaints, so you can keep an eye out on that.
There are also some best practices to keep in mind when sending system emails like this, like not using that same system for marketing emails since they're more likely to get flagged as spam.
They do, although AFAIK they recommend using the API. I recently moved from one to another and it only took switching out the library and a few lines of code, since most of them work very similarly.
There are also some best practices to keep in mind when sending system emails like this, like not using that same system for marketing emails since they're more likely to get flagged as spam.