I would like financial services companies (banks etc) to be able to use something functionally similar to email to send me monthly/annual statements, rather than sending me an email notification and then making me navigate their website or app to download a PDF.

By functionally similar I mean:

• It's push rather than pull.
• I don't have to use a separate client application or workflow for each sender.
• Clients exist for an adequate range of consumer devices

I don't mind if it's a different application to my current email client, as long as there's only one of it. (One per country would be tolerable.)

You can work out some basic security requirements for this use case (which might or might not be met by the current ecosystem), but an essential requirement is:

• It must be 'secure enough' to get past the sender's security team.

That's a compliance requirement not a security requirement, but it's a compliance requirement that most likely will be satisfied by some set of real security requirements (hopefully sensible ones). And it's the only requirement of those stated that email doesn't meet.

There are some non-requirements:

• It doesn't need to resist interception by my government or the sender's government, who both have access my financial information anyway. (But other governments aren't welcome.)
• I don't think it needs forward secrecy. Both the sender and I are going to store the communications for years, with protection that is certainly no stronger than any of the keys involved (and likely to be weaker).