Do what works for you of course, but you could also just run your makefile on our PC and then push the generated HTML to the gh-pages branch/repo. I don't see (much? any?) value in running that on a server, if it's not triggered regularly for some external source. There are no simpler tools here, just the same with the gh layer on top.
The value is 1) I can edit the site from any device with a web browser, 2) I don't want to run a process manually, I want it to be automatic (I don't want extra work every time I change a line of text), 3) other people can contribute content through a PR and everything is fixed again with no need for me (or them) to run anything, 4) there are canned github actions to do things so I don't have to figure them out, 5) once I add the Pages plumbing it'll automatically publish a website for me
I don't get 4) and 5). Plumbing for what? This makes me think it is not understood that github pages will just host any html you put in its repo. There is no magic plumbing in play here, not on your side at least.
1) okay, I can see that (also @simonw, the other comment). 2) is a tradeoff, you have a lot more work with setting this all up. but okay. For 3), if building the site is easy enough and documented properly, having to run a build script once and having it as part of the PR is in my experience not a problem for contributors - and there is also a tradeoff here against them having to understand what hidden steps the Github action will do, when wanting to contribute.
