I'm not using a private CA for my internal services, just plain self-signed certs. But if I had to, I would probably go as simple as possible in the first time: generate the CA cert using ansible, use ansible to automate signing of all my certs by the CA cert. The openssl_* modules make this easy enough. This is not very different from my current self-signed setup, the benefit is that I'd only have to trust a single CA certificate/bypass a single certificate warning, instead of getting a warning for every single certificate/domain.

If I wanted to rotate certificates frequently, I'd look into setting up an ACME server like [1], and point mod_md or certbot to it, instead of the default letsencrypt endpoint.

This still does not solve the problem of how to get your clients to trust your private CA. There are dozens of different mechanisms to get a certificate into the trust store. On Linux machines this is easy enough (add the CA cert to /usr/local/share/ca-certificates/*.crt, run update-ca-certificates), but other operating systems use different methods (ever tried adding a custom CA cert on Android? it's painful. Do other OS even allow it?). Then some apps (Web browsers for example) use their own CA cert store, which is different from the OS... What about clients you don't have admin access to? etc.

So for simplicity's sake, if I really wanted valid certs for my internal services, I'd use subdomains of an actual, purchased (more like renting...) domain name (e.g. service-name.internal.example.org), and get the certs from Let's Encrypt (using DNS challenge, or HTTP challenge on a public-facing server and sync the certificates to the actual servers that needs them). It's not ideal, but still better than the certificate racket system we had before Let's Encrypt.