How can I get email to "me" to email to my email address? If I tell Siri "email me," the right thing happens: a new message is created with my email address as the recipient. But if I open the email app and type "me" as the recipient, I'm told "Invalid Address" when I try to send the message.

How can I get the email app to understand that "me" as a recipient means to use the email address corresponding to who I've told Siri is me?
You can't.
Because of the design of iOS, there is no native way to accomplish this. The reason for this is that the address field in mail and messages are almost exactly identical, not just in looks, but in function.
In messages, you can actually send a text to me, but it will send whatever you said straight to 63 (see number pad). This is because some companies hand out shortcodes, and say something like Text APP to WMART (96278) to download the Walmart app!. users can either punch in 96278 into messages, or they can punch in the much more memorable WMART into the address field and still have the message go back to the right place.
Now, correct me if I'm wrong, but I believe that mail will actually try to send an email to me if you hit send. This may be a planned feature for within organizations or something, or it could be an oversight. But due to the design of iOS, you are unable to accomplish what you wanted.