Rather than replacing your Makefile with GH Actions, replace your GH Actions with a Makefile, and make your GH Actions run `make` in a script task.
Do you really need that GH Action for pulling Docker images / installing $language_compiler / creating cloud resources ? A `docker` / `curl` / `sudo apt-get install` invocation in a Makefile / script needs to be written once and is the same in CI as on your dev machines. The Action is GH-specific, requires you to look it up, requires you to learn the DSL for invoking it, requires you to keep it up-to-date, requires you to worry about it getting hacked and stealing your secrets, ...
A Makefile already supports dependencies between targets. A shell script is a simple DSL for executing processes, piping the output of one to another, and detecting and acting on failure of subcommands. How much YAML spaghetti do you need to write to do those same things in a workflow file?