Level 1) you package your service into a zip/rpm/deb/etc and have an agent on the machine that periodically pulls
Level 2) you pack your software into an ami and use the update the asg config. You can periodically "drain" the asg of old instances
Level 3) you deploy your stack again with the new stack having the ami that you've build at level 2 referenced. You start shifting traffic between the old stack and the new stack. You monitor and rollback if something is wrong.
I find it's easier to use Ansible/Salt/Puppet Bolt and Packer to bake an AMI every night, update the launch template in a DB (which Terraform pulls the value from, thus there is no drift), and auto the ASG. Then you just force a drain.
Now you've got automatic, constantly updating VMs every night if you want them. And a new deployment is just commiting code to master and pushing and that whole pipeline triggers for you.
People like to overcomplicate things, Mirceal. You're on the right path :-)
I'll be honest I haven't fully explored AMIs as a solution but how do you run the AMI in your local dev environment? I can replicate the same K8s with docker images easily in local dev.