What developers tend to do when forced to use immutable infra is to move configuration from on-disk files to RAM and query it using network API from a central system. The problem is that it makes systems less reliable. If a VM/server restarted it can practically always read a config form disk but if you service relies on an external system to get runtime configuration it would not work if this external system is down, overloaded, misconfigured, returns wrong config because of a bug e. t. c. And it does happen in practice even when system designers tell they configuration API is very reliable (in theory). After seeing such systems fail I like simple on-disk configs more and more.
CM is useful here. And usually in the bootstrapping of such architectures.
Doesn't that just move where you need CM a bit?