Yes that will be great feature. But you would want that to happen only when you select the recurring invoices to be generated when you go to invoices and have the notification on top and click view. Then only should it mail automatically. For the case like rent income. If someone no longer rents with you, you can choose not to generate that invoice and delete the recurring invoice after.
I agree, it would be ideal for these invoices to go out automatically. In my organisation we are all volunteers and any time spent sending out invoices is taking away from other more useful community activities that we could be doing. Also we might not log in for a week, and realise that invoices need to be sent, and send them out a week late. Both Xero and Kashflow will automatically send out recurring invoices.