And this is different from DNS how exactly? The key and resulting cert still needs to be distributed among your servers no matter which method is used.

With dns-01, multiple servers could, independently of each other, fetch a certificate for the same set of hostnames. Not sure if it’s a good idea though.

Multiple keys and certs for the same hostname? Will the CA even issue that?

I guess it depends on the CA, but some do. Let’s Encrypt does, for example. I guess it’s useful for HA deployments, where load balancers might be spread out across multiple datacenters and stuff like that.

NB that rate limits apply https://letsencrypt.org/docs/rate-limits/

Not really, just forward .well-known/acme-challenge/* requests to a single server or otherwise make sure that the challenge responses are served from all instances.