How much is this true?

Not only are those clouds expensive, they are also slow.

So perhaps a fifth of the peak hardware would be cheaper than the entire AWD and still more than capable.

Operating and administering your own systems infrastructure requires that your business invest in the people to do so, this builds institutional knowledge which makes the important bit, the data and knowledge, portable. If the VM in the cloud gets too expensive you can use another provider, or you can buy hardware and run it locally. If the VM provider cuts your service you still have access to your data because you never lost control of it. Problems can be fixed by in house staff that don't suddenly evaporate for arbitrary reasons or have service outages.

If your entire business depends on Microsoft services and it gets too expensive you have no options but to pay more. If your account gets locked then you're out of business until you can get Microsoft to give you access again. If you want to migrate away, there isn't another Microsoft to move your data to and you've replaced all of your technical staff with a support phone number, which isn't currently accepting your calls.

It's a VM that you set up, you have the image yourself, you could put it on a machine in your living room if you had to.
"I'm paying for a colocation of a machine I administer" is very different from "I've written my application such that it can only run inside an AWS system"

The idea is that your services run on remote systems without regard for what those systems are (as a VM, docker image, etc.) Your architecture is decoupled from theirs - you can run on an Amazon host one week, and a server in your closet the next.

And as a bonus, systems hosted this way are often harder to scrape as they're all structured differently. Additionally, you can (and should!) take additional measures to protect your data from your provider - something that just can't be done when the provider controls the data architecture.

The end result is the same:
You control what the machine does. The data as well as backups (assuming you arent using specific hardware offerings but just compute and storage)

Example:
I am done with AWS pricing and Azure gave me a fat stack credits to go over there.
Agnostic VMs could be backed up and migrated over to Azure.
Essentially the same as migrating Hyper-V or VMware to Proxmox-VE

Solid example. I could pay Lyrasis to host an instance of Archivesspace for me. They'd control updates, backups, etc, I'd just use the web interface to manage my archival collections.

OR I could rent a server, install Archivesspace myself (it's open source), sysadmin it myself, take on all that headache and control.

They're both in the cloud, but one's software as a service (SAaS) and the other is just a Linux box on someone else's machine. The second is cheaper in my experience, but only if you have someone that can sysadmin it. Otherwise you've got a learning curve ahead of you.

(it's late, so feel free to tell me I've misread the thread).