Are there serious problems with an EC2 auto-scaling AMI that automatically downloads newest server from git?
What you're going to do is only calling for inconsistencies between your nodes.
I wouldn't do any deployment during the instance boot-up, instead install AWS CodeDeploy as part of your boot-up (or bake it into the AMI) and once the instance boots up it contacts the CodeDeploy service and obtains the newest code version.
Likewise if you need to roll-out a newer code version push it to CodeDeploy and it will take care of updating all the servers in your Auto-Scaling Group in a consistent, predictable way.
With ASG you really really want to prevent inconsistencies between the nodes and this is one of the ways to do it.
Hope that helps :)