Update to my question: Someone wrote this blog post about this exact problem, even with this exact extension (dated September 25, 2018), and how to manually update the extension: https://www.ghacks.net/2018/09/25/how-to-update-chrome-exten...

UPDATE

After reading the comments on the linked blog article, frantically hitting the "Update" button (in developer mode) many times indeed helps! I had clicked it quite a few times already, but "normally", going crazy and hitting it a lot and quickly indeed lead to the update occurring.

Wow, what a crazy user interface that requires users to go berserk on the buttons...