You probably haven’t seen any “wide gamut” demos with fundamentally different primaries involved; display manufacturers have been expanding their gamuts by making the three RGB primaries have narrower and narrower spectral power distributions, rather than by adding additional primaries.
For a while Sharp made a “4-primary” display, but the additional “yellow” primary was more or less created by an additive mixture of existing “R” and “G” spectra, so didn’t really make as much practical difference as one would hope. https://en.wikipedia.org/wiki/Quattron
You can get a display with like 8 or 10 primaries if you run a color science research lab and can spend tens of thousands of dollars on it (or maybe more? dunno).
If you look at a photo produced by an RGB digital camera, color graded in some standard RGB space, and then presented on an RGB display vs. a 10-primary display, you aren’t going to see any obvious difference, because the content does not take any special advantage of the medium.
In comparison, if you look at a bunch of synthetic images created to highlight the differences / stretch the capabilities of the displays, or e.g. a video produced by a whole hardware–software–author pipeline expecting the more capable display, the differences will be pretty obvious.
If you want a somewhat relevant test, you can try comparing images produced by a photo printer with images printed in old newspapers. Having a larger gamut makes the former much more vibrant.
Current displays produce very intense orangish-red, purplish-blue, yellowish-green, and reddish-purple colors. So the differences with a hypothetical >5-primary display are going to be most noticeable in the blue-green range, and oranges and yellows.
If all you're paying attention to is what's on the screen then it probably doesn't matter much exactly what those colours are because it's all relative, your brain compensates for it, and it's not as if the cells in your eye have a consistent response to colour in any case: it all depends on what you've been looking at recently.
If you've ever been in a television shop, back when they still had shops, you could find two screens next to each other showing the same television channel but with radically different colours, when you compare them side-by-side, but if you watch just one of the screens - it doesn't matter which - you get used to the colours of that screen and everything looks fine.