I'm guessing someone out there's thinking: Why aren't they hosting in the cloud? The cloud being either Amazon or Azure. Surely nothing else exists. Is it really possible to host your own PHYSICAL machine? Does that count as the cloud?!
In all seriousness, however, the decision (likely) has very little to do with that. They're most likely not hosting in the cloud because the current CA/Browser Forum rules around the operation of public CAs effectively don't permit cloud hosting. That's a work in progress, but for the time being, the actual CA infrastructure can't be hosted in the cloud due to security and auditability requirements.
I can forsee letsencrypt in the future going to building their own cloud (on their own physical infrastructure), but speaking as a letsencrypt user of their free certificate program, I would lose respect and interest in their service if they went with an AWS or GCP or Azure approach.
The independence from other major players (and the ability of their team to change and move everything about their service, as needed) is one of the reasons I use letsencrypt.
So long as they don't have a viable independent revenue stream they're arguably less independent than commercial CAs.
For example : https://www.scaleway.com/en/dedibox/dedirack/
I'm not sure you can say it's the cloud though.
They are not hosting their database in the cloud like Amazon or Azure because no cloud provider offers such high performances at a comparable price. Actually I'm not even sure you can get a cloud VM with that many IOs, if you don't mind the pricing.
It can help if someone wants their data "in the cloud".