Migadu is relatively private, dirt cheap, and dead easy to set up. Supports both web and desktop email clients of choice.

IMHO, TLD matters and always will - the problem is that it matters to varying degrees depending on the destination host, the remainder of your setup, etc. If you're fastidious about configuring all of the requisite DNS records, etc., it will be less of a problem.

I mostly avoid the newer non-CC TLDs for that reason, and general personal preference. Deliverability is enough of a challenge without adding more work to it, which is why 'actually' hosting one's own email is mostly a toy project, and not something generally done as a serious endeavor. Useful for learning and understanding, of course, but not particularly practical to literally host one's own email server for ongoing usage in any critical use case.

You'll find certain providers easier to get your emails through to than others - Hotmail and variants are notoriously difficult, and tend to drop inbound mail at the gate without sending a bounce message, as if the inbound mail just disappeared.