Indeed, and that is what we're doing right now. However, it would be nice if they dealt with creating nicely formatted invoices and emailed them so we don't have to.
Where does that end? They're taking your payments, making your invoices, why not be an OAuth provider for your users? Host your app? Store and ship your physical products?
I love Stripe. They do one thing well, better than anyone else I have used. It's not perfect for everyone, but anytime someone tries to be perfect for everyone they stop being perfect for anyone.
I don't think it's that crazy to wish for emailed invoices from a payment processor. Unlike the examples you provide, it's a pretty natural and easy add-on to such a service which many providers (eg, PayPal, Square) do.
PayPal and Square are user facing companies. Stripe is currently strictly a developer facing company (I mean look at their home page, they have code samples!).
Stripe provides hooks that allow you to easily send your own invoices--that's a developer friendly feature that maintains flexibility (everyone will want invoices to work differently).