I don't know nor use the app promoted by GP, but in general the problem with apple is that they don't support the standard email/IMAP push, and only some 3rd parties have approval to use the non-standard apple email push (Fastmail I think). So people have come up with ways to get around apple's arbitrary restrictions.
My dad has been using his own domain for email for about 25 years and has bounced around to various hosts... random hosting providers. He uses Apple Mail and hasn't ever mentioned having an issue. Though I'm unsure if he uses push of fetch. I guess I always assumed he was using push, but maybe not.
Of course we're all hoping that clients move to JMAP at some point, where push is first-class citizen and we don't need the very custom hack that Apple uses to receive pushes right now.
I've been using the Fastmail app to get started, which has been working great and the migration tools worked really well. Once I'm fully migrated I may want to set up the native apps for some added OS integration.
Local search should be single digit ms, for example.