I solved it finally.
The problem is that when you download the Couchbase Lite ZIP distribution the support native libraries are located inside folders named support/linux/x86_64/libicu, support/linux/x86_64/libc++ and support/linux/x86_64/libz. I just copied the whole support folder to my project and added LD_LIBRARY_PATH pointing to support/linux/x86_64, trying to follow this snippet from the docs:
export LD_LIBRARY_PATH=<pathToCbl>/support/linux/x86_64/:$LD_LIBRARY_PATH
But no, it doesn’t work that way. The JVM when linking to the native libraries won’t go and look in the subfolders. So the fix is simply moving the contents of these three folders into /support/linux/x86_64/. Now Github Actions are happy and so am I.
For more experienced Linux devs I suppose this is obvious, but every day you learn something new. 
Could I suggest rewording the forementioned documentation to make this more clear? Thanks!