[tialaramex]

Ah not quite, I think pooling would be rather easier than you've thought, there are Let's Encrypt people here, but let me explain what you'd do to have N unrelated machines which are all able to successfully claim they are some-shared-name.example

Each such machine mints (as often as it wants, but at least once) a document called a Certificate Signing Request. This is a signed (thus cannot be forged) document but it's public (so it needn't be confidential) and it basically says "Here's my public key, I claim I am some-shared-name.example, and I've signed this document with my private key so you can tell it was me who made it".

The centralized service collects these public documents for legitimate members of the pool and it asks a CA to issue certificates for them. The CA wants a CSR, that's literally what it asks for -- Let's Encrypt clients actually just make one for you automatically, they still need one. Then the certificates are likewise public documents and can be just provided to anybody who wants them (including the NTP pool servers they're actually for which can collect a current certificate periodically).

So you're only moving two public, signed, documents, which isn't hard to get right, you should indeed probably do this out-of-band but you aren't sharing the valuable private key anywhere, that's a terrible idea as well as being hard to do correctly it's just unnecessary.