What reasons should I NOT use a private DNS zone and CNAME services in my AWS/K8s environment?
It seems a reasonable idea to me.
There migt be a few things you might want to check tho :
This article or this one. For example :

the fact the that a private hosted zone is associated with a VPC
there is some DNS hostnames and DNS resolution parameters on the VPC settings


enableDnsHostnames : Indicates whether instances with public IP addresses get corresponding public DNS hostnames
enableDnsSupport : Indicates whether the DNS resolution is supported.


Only VPC resolver will resolve the private records (subnet range +2 for the ip of the VPC resolver, search for reserved ip here)

take care you didn't configure custom resolver and so wont be able to access your private records
think about the fact that you won't be able to resolve those records from outside the VPC (office for example) unless you manage to plug to the VPC resolver somehow



You could also give a look at Alias record ?
Could give advantages like :

create record that has the same name as the hosted zone
not being charged for the queries (in opposite to CNAMEs)

These are few things I can think off right now. Hope this helps.
Cheers.