Using Gentoo stable in production right now. I'm in charge of how long a package is supported now. All execs get a brand new gentoo machine built with binaries compiled by myself.
You wouldn't believe how fast you can get a gentoo machine up and running compared to other distros. Build for a minimum common architechture (all intel binaries are based on Sandy Bridge, all ARM based on Rockchip RK3088), and installing for new computer is little more than untarring a bunch of binaries to /. My record is 5 minutes for a full KDE Plasma 5.5 software stack.
I explicitly did not mention Gentoo - I know a bunch of people who run it in production. But, for anyone considering to do this - if you're running Gentoo, you're essentially building your own distro, which has massive advantages but is also a huge effort. You're now in charge of security updates, maintenance and Q&A. What if you're leaving the company? There are many Debian or Redhat admins, but good luck finding a Gentoo expert.
We use Alpine Linux for our applications and I like it, and I too shudder at it being used for the entire production system. As a sysadmin, you can still administer the LTS distro that hosts the docker containers and whatever other pieces of the stack you interact with. Alpine Linux containers, like any other container, should host an instance of an application (maybe not even that, depending on how complex the application is) -- not the entire production server, not SSH keys, not iptables, firewall rules, etc.
The value that stable, long term support distros provide shouldn't be underestimated.