Whether you are paying for HA depends on your Recovery Time Objective (RTO). You can have a bunch of suspended EC2 instances and non EC2 resources where you only pay per use in another region.
You can redirect traffic to another region and have autoscaling spin up EC2 instances, etc.
Sure, if you can wait for it to load from unallocated resources (and risk failure) then it's a very different scenario.
But, very notably, you can have a suspended cloud backup even if your main servers aren't cloud. And the added complexity for datacenter-to-cloud HA doesn't have to be significantly higher than the cloud-to-cloud version.
You can redirect traffic to another region and have autoscaling spin up EC2 instances, etc.