What overhead?
Just about every web server these days supports ACME -- some natively, some via scripts, and you can set up your own internal CA using something like step-ca that speaks ACME if you don't want your certs going out to the transparency log.
The last few companies I've worked at had no http behind the scenes -- everything, including service-to-service communications was handled via https. It's a hard requirement for just about everything financial, healthcare, and sensitive these days.
> What overhead?
[proceeds to describe a bunch of new infrastructure and automation you need to setup and monitor]
So when ACME breaks - which it will, because it's not foolproof - the server securely hosting the cafeteria menus is now inaccessible, instead of being susceptible to interception or modification in transit. Because the guy that has owned your core switches is most concerned that everyone will be eating taco salad every day.