How does using packer and terraform in a CD pipeline compare to docker images built from git?
You're comparing apples with oranges according to your description. A strict comparison could be packer toward dockerfile, and/or terraform toward kubernetes yaml for example.

Main difference between your docker pipeline and your packer/terraform one is the architecture behind them.
What is putting the burden here is that your AWS infrastructure doesn't take advantage of auto-scaling groups rolling updates, or your 'stage-gates' in go does disrupt them.

You're not comparing docker with terraform, you're comparing rancher with your AWS infrastructure choices. If you don't use the equivalent bricks of rancher load balancing and deployment groups (kubernetes wording here) which are Elastic Load Balancers and Auto Scaling Groups in AWS you can't compare them.
That's saying a car and an bicycle does the same thing (transportation) with a better result for the car, put a motor on the bicycle and you'll have a far different appreciation and something less biased.

Your observation is pointing out that your usage of AWS facilities is sub optimal, terraform (or whatever provisionning system) can't help you more than running things, if your architecture is not made (in the terraform file) to allow rolling updates from this facility, no automatic system will fix it for you as you'll have to code it again (test instances status, etc.).

Of course rancher pre-bake most of those, making them invisible to you. This also mean you are enforced in rancher methodology (well, swarm, kubernetes or messos architecture to be exact).
That's the main difference, on one side you have an infrastructure already ready to serve with rancher and you can't run outside it, and on the other side you have to think your architecture before hand. If you were to run docker containers on a bare docker host (without any orchestrator on top) you'll end up with the same architecture design to think before hand and the same caveats you're seeing on AWS.

AWS is not strict on how you use it, it's an infrastructure as a service with no opinion on how you use it, whereas rancher enforce a bunch of services around your own container to save you the design time as long as you accept the constraint within it.

Hope this help seeing the problem from an upper level and as such I think the answer to your questions is that you were missing a pattern ;)

For the records: we're doing deployments on AWS with cloudformation (native version of terraform, with its own caveats) from  a push on a gitlab server and it works nicely, the work to get proper cloudformation templates and the script to run it was non negligible. This to say the tools are not the problem, it's all a design choice at a point, same results can be achieved, arncher comes with its own caveats and design choices to make when you integrate it into an existing infrastructure.