docker based workflow cannot cache

Describe the bug
For setup-* actions they cache tools in RUNNER_TOOL_CACHE.
Doing so does not seems possible in docker based actions although the env variable is available.
For example on ubuntu github hosted runner, RUNNER_TOOL_CACHE has the path of /opt/hostedtoolcache which is not mounted onto the container, thus making it impossible to add a tool to cache and add it to path.

I tried other solutions by caching it in GITHUB_WORKSPACE but paths are not translated when being written to GITHUB_PATH, also the workspace may be the repository if checkout has been it before this action.

Expected behavior
I would expect the container workflow to have environment variables leading to usable directory and potentially paths to be converted when written in GITHUB_PATH as paths in env variables does not correspond to the runner ones.

Runner Version and Platform

Current runner version: '2.303.0'
Operating System
  Ubuntu
  22.04.2
  LTS
Runner Image
  Image: ubuntu-22.04
  Version: 20230402.1
  Included Software: https://github.com/actions/runner-images/blob/ubuntu22/20230402.1/images/linux/Ubuntu2204-Readme.md
  Image Release: https://github.com/actions/runner-images/releases/tag/ubuntu22%2F20230402.1
Runner Image Provisioner
  2.0.139.1

No source file or test is named. Start by reproducing the issue with a Docker-based action on the stated Ubuntu runner, then trace how RUNNER_TOOL_CACHE and GITHUB_PATH are exposed and processed; done means setup-* tools can be cached and added to PATH with usable container paths.