Use mailman.
Get one list for "newsletter@yourdomain.com". Put your people on it.
Configure it so that only you can post to it. You don't need a public archive, but you have that option. Keep membership-listing off. Make sure each outbound message has the option to self-unsubscribe. Your public page can even have the functionality to let people self-subscribe (or self-resubscribe). This is a default, I think.
Choose a mailman host with a track record of many years, e.g. over one decade. That knows how to manage mail. That knows how how SPF/DKIM/DMARK and discoverability works. That knows how to have good IPs. For example, a host that has managed reputable open-source mailing lists on the long term, has more than one person running it, etc. You have your choice of hosts anyway. If one goes down, there are others to choose from. That should not be a big deal.
When you want to send your people a newsletter, just send an email to newsletter@yourdomain.com.
I could not imagine anything simpler, easier, or most cost effective, for what you ask.
