Hi,
You must be misunderstanding what is recurring payments.
Recurring payments means that once the user validates the payment on paypal, each period, the funds is taken automatically from the user account to yours by PayPal and the subscription period is extended automatically without any interaction. The user has to cancel the recurring payment on his paypal account in order to stop the renewal.
The current integration with akeeba subscriptions does not handle that. What it handles is the normal subscription mode, like you're doing at the moment with acymailing: when the subscription period reaches the end, an email is sent to the user so that he renew by buying again the product.
So from what I understand, what you want to do is already possible.