wintermute.: It shouldn't be that hard to design a page with HTML5 which also works with an Firefox fork like Palemoon ...
And that exactly is the problem. HTML 5 is deprecated. The current W3C recommendation is HTML 5.2. HTML 5.3 also already exists as working draft.
Palemoon on the other hand uses the Goanna engine (which is a fork of Gecko, the old Firefox engine). According to Wikipedia it supports "HTML versions 3 and 4, and most of the living HTML5 standard." In other words it isn't even fully HTML 5 compatible, not to mention 5.1 or 5.2 compatible.
While Palemoon does get backported security fixes the featureset is frozen as far as I know. Which means that featurewise Palemoon has been outdated for at least 3-5 years. Which means that as time goes on you will find that more and more pages don't work properly with Palemoon.
GOG has already given an answer to the question how they intend to handle this problem: They don't have the manpower to test their page on every possible version of every possible browser so they officially only support the latest version of the most common browsers. If their page runs on that they don't consider it an error and won't fix it. If it doesn't run on the browser of your choice then update or change your browser.
While I don't like this answer either and would also prefer to use a different browser I also understand that a small company can't possibly write and maintain X versions of their homepage for X different browsers. So my solution to this problem was to install the portable version of the latest Firefox as secondary browser that I use for pages that don't work right in older browsers. You might want to do the same.