Nothing stops you from doing infra as code on dedicated hosting. All of my setups for the last decade or so have been built around VMs and/or containers with deployment systems where we spun up containers across multiple datacenters on servers we had full control over.
Many dedicated hosting providers now provide APIs for deploying servers, as well, so you can handle even deployment of the underlying servers in an automated way.
Several have combined cloud deployments with deployments to dedicated servers from the same container images, bound together in a single virtual network. E.g. I had client that hosted across AWS, GCP and Hetzner, and migrated services between them zero-downtime. Eventually they moved everything to Hetzner because it cost them about 1/10th of AWS and GCP given their bandwidth use (at the time outbound bandwidth at AWS cost 50x what it cost at Hetzner).
If organizational dysfunction means you're not allowed to order the resources you need, then that is of course a problem, but a very different one.
Many dedicated hosting providers now provide APIs for deploying servers, as well, so you can handle even deployment of the underlying servers in an automated way.
Several have combined cloud deployments with deployments to dedicated servers from the same container images, bound together in a single virtual network. E.g. I had client that hosted across AWS, GCP and Hetzner, and migrated services between them zero-downtime. Eventually they moved everything to Hetzner because it cost them about 1/10th of AWS and GCP given their bandwidth use (at the time outbound bandwidth at AWS cost 50x what it cost at Hetzner).
If organizational dysfunction means you're not allowed to order the resources you need, then that is of course a problem, but a very different one.