Introduce them to the idea of having something like Caddy sit in front of apps solely for the purpose of doing TLS termination... Caddy et al can update the certs automatically.
Giving the TLS endpoint itself the authority to manage certificates kind of weakens the usefulness of rotating certificates in the first place. You probably don't let your external facing authoritative DNS servers near zone key material, so there's no reason to let the external load balancers rotate certificates.
Where I have used F5 there was never any problem letting the backend configuration system do the rotation and upload of certificates together with every other piece of configuration that is needed for day to day operations.
Or request a certificate over the public internet, for an internal service. Your hostname must be exposed to the web and will be publicly visible in transparency reports.
Key loss on one of those is like a takeover of an entire chunk of hostnames. Really opens you up.
That doesn't seem like the end of the world. It means you shouldn't have `secret-plans-for-world-takeover.example.com`, but it's already the case that secret projects should use opaque codenames. Most internal domain names would not actually leak any information of value.