You can use actual CH Cloud on AWS?..

Do any two teams actually operate it in anything like the same way though?

What I saw of it, especially some years ago, was it was highly particular, and everyone had their own odd habits built around running it, ingestion, querying, everything, to the point I suspect there are a non trivial number of companies using it where it is actually the core operational expertise of the company, despite them all appearing to be in totally different domains.

You can use ClickHouse cloud to host it on AWS.

But given the majority of use-cases of CH, AWS can be quite expensive.