Secure and private mail client for iOS

In no particular order...

Among the available mail clients for iPhone, there is the built-in mail client, and other third-party mail clients such as Spark and Edison are available in the app store. I haven't gone looking for FOSS mail clients.

The built-in mail client works fine with POP/IMAP/SMTP/ESMTP mail servers, and with Microsoft Exchange Server.

Login credentials can be stored locally, or can be stored in iCloud Keychain with access to the latter limited to the user's own devices.

The Apple mail client is MIME-encoded HTML, and not plain text.

I'd suggest looking at PinePhone or PinePhone Pro here. Those products will fit better with what I'm inferring to be your general preferences.

More generally, Apple has opinions on how things should fit together and work. If those opinions should match your needs and expectations, great. If you should want or need to tailor or tinker, that can become a darn hard slog.

Outlook, Edison Mail, Proton. Those are the first that come up in a search of the app store. But I think the built-in mail meets your requirements also.

In what way does the native mail client not meet these requirements? Or Outlook for that matter?

The first item on your list is largely going to be a function of the mail server, not the client. All the client has to do is make a secure connection.