Integrating Heroku System Errors with Sentry

I want to integrate Heroku router and system errors using Sentry. The error logging of my application with Sentry is great, but I am missing the application timeouts and crashes that occur outside the application from the Heroku system. There are logging tools that provide all Heroku syslogs, but I would rather use a watch device as the only tool for all the checks and error notifications.

As far as I can see, this is currently not supported via regular Sentry / Heroku. The only way I can see to achieve this is to create your own custom log drain that will filter heroku logs and forward what I consider to be "bugs" to Sentry. Am I on the right track, or am I missing some part of the ecosystem?