When I check github.com, there is a newer version (1.0.2).
That is incorrect. If you look again, the versions on the Releases page look like 1.0.2-rc2. The "rc" in that version number stands for "release candidate" these are pre-releases intended to be used by beta testers. Releasing pre-release versions on the unsuspecting users who have no idea what -rc means and will just click "Update" causes a huge amount of problems. This is why the authors of the ESP32 core have created a separate "development" Boards Manager URL to be used by people who want to do beta testing. It's listed on this page:
You can simply replace your existing URL in File > Preferences with the development one, after which you will find the -rc versions available for installation in Boards Manager.
Keep in mind that pre-release versions are not as thoroughly tested and may have issues. The whole point of pre-releases is to allow people who want to help out with the development process to test it out and find the bugs so they can be fixed before the production release is made.
Is there a systematic way I can use/test the beta version directly from github?
Yes. Follow the instructions for your OS under "Using Arduino IDE with the development repository" here: