If I was relying on a third party for push notifications (something I can easily do on my own, btw), the one single reason for downtime that would totally not be acceptable for me is them forgetting to renew their domain.

This is a huge indicator, IMHO, that they don't really believe in their service and don't even do the absolute minimum that's required for keeping it up.

Worse, depending on the type of message I'm passing through them, this might have privacy or security implications as messages will now be sent to a non-related third party.

Stuff like this is why I personally am very, very careful before outsourcing any part of my core infrastructure.