"Getting a push alert any time there is an error helps us respond to issues as soon as they come up. "
Oh god no.
Maybe if this system is not expected to ever grow in complexity. Otherwise this will become a big source of frustration really soon. And even if it does not, it's very easy to get a flood of non-actionable alerts for a transient condition.
It would be better to collect error rates and alert on that. Also the other golden signals.