Good question. I’m not sure how you’d bring the GH data into AWS, but there probably is some way.
The mean time to recovery and similar metrics don’t have anything to do with GitHub because they measure what’s happening on AWS itself, so you’re good there.
For the others, maybe you could look into publishing custom CloudWatch metrics: https://docs.aws.amazon.com/AmazonCloudWatch/latest/monitoring/publishingMetrics.html
Maybe there is a better way and I’m not sure what the state of the art is. But I would think it shouldn’t be too difficult to trigger calls to AWS to publish a metric on pushes, merged, etc.
Are you also using GitHub Actions for building?