Sounds like FUD to me. These neither of these arguments apply specifically to docker registries and are more an argument against any form of self-hosting, which is a totally different conversation.

Having to rebuild if your server dies and you have no backups is obvious, same is for true for any part of your infrastructure and seems more like a deeply flawed ops strategy as a whole at the organization.

And in this case we are talking about self hosting as in not buying it as a service, not necessarily on your own hardware. I've self-hosted docker registries on a VPS from cloud provides who also provide backup and resilience against underlying hardware failures as a part of their service. If a server dies the VM is transparently moved to another machine, no big deal.