Source-based distros essentially do that, they just cache the binaries.

Various research OSs are JIT-based, of course. It looks like JX (a Java operating system) caches its native code, so it's not 'pure JIT' https://github.com/mczero80/jx/blob/5fbeae79/libs/compiler_e...

It looks like Cosmos (a C# operating system) does the same https://en.wikipedia.org/wiki/IL2CPU