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.
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.
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).
Stripe is a payment processor. If you want invoicing, use (FreshBooks|Xero|Blinksale|etc).
"It would be nice if" is often the death of good product design.