Ok, if you’re unable to install a version of macOS that’s newer than the 7-year-old macOS 10.13, then unfortunately you won’t be able to use the up-to-date major browsers.
Your site and theme are indeed working properly, but they are both using more modern web technologies than your browsers can support. Similarly, over time, you’ll begin to see more display anomalies across the web.
I do have good news for you though. https://icab.de has a reputation as the old Mac workhorse of the web. It was one of the first Mac-only browsers, it’s still being developed today, and its latest release (6.2.3, released July of this year) requires at least macOS 10.13 according to their download page. I installed it and can confirm that it is displaying your test page properly. iCab can be behind modern web technologies at times, so I generally don’t recommend it over up-to-date major browsers, but you might want to consider switching to it until you get a Mac that can support newer versions of macOS (currently, Firefox requires macOS 10.15 or higher, and Chrome requires macOS 11 or higher).