Even better to use someone that isn't your payment processor, so that should you need to change payment processors you don't also have to re-acquire all the billing info from your customers. You can use Stripe today, PayPal tomorrow, and Braintree the next if that's what works best for your business.
There's always going to be single points of failure, but which is more likely: you want or have to change payment processors (you've been terminated, your fees have gone up, you want to switch to a lower cost provider) or you want to change flat-rate vaulting services? Plus, Spreedly will give you your data if you leave, whereas there is no way to get stored billing info out of most payment processors.
Card vaulting as a service: https://spreedly.com/ ($10/mo for up to 5000 cards)