Even if you're not all-in on AWS, you can use Parameter Store for a trivial sum. The latency might be a bit higher, but in most cases it shouldn't matter since secrets usually have a nonzero TTL associated with them.
If you are using AWS, EC2 itself is a trusted third party that grants the appropriate permissions for your executing system (container, instance, whatever).
If you do nothing you end up stuck with secrets in git history and you essentially have to roll them all.
So, my advice is generally SSM with no security is better than no secrets Management solution.
SSM wins for mine. I think you're right that the file approach wins for yours.
To that end, yes, their servers are probably hand-rolled or provisioned with minimal scripting like Ansible. And so manual secret deployment is to be expected.
If somebody is attempting to autoscale without having this solved already, they are making mistakes.