I think it's fair to say that, as far as commercially available consumer computer displays are concerned (and thus anything relevant to anyone viewing the OP website), we are indeed display-bound.
And given consumer media are RGB encoded (or any triplet of colour flavour), it will be long before having consumer displays with more than 3 usable bands. Sharp Electronics had a display with RGBY(ellow) filters, i.e. Quattron (https://en.wikipedia.org/wiki/Quattron), but no media existed with the 4 required channels, not only that but there was no Yellow primary thus it was "useless".
You don't need a 4th primary to represent all visible colors. You just need imaginary primaries. Heck, even something with real primaries like Rec. 2020 [1] can represent information outside the standard sRGB primaries that a 4th primary would be useful to display. (Granted, a better choice of green primary would work better in that specific case, but such choices aren't always physically realizable.)
Another use is enhancing saturation of an image. My DLP projector has a mode like this to take advantage of its 6 primaries. Enhancing saturation can produce a convex gamut even from a non-convex one (e.g. sRGB), the only means of reproducing of which may be with additional primaries.
By my estimation, the main problem with Quattron wasn't that it had a 4th primary, which is legitimately useful. It's that - as you point out - it didn't. It literally did not have a yellow primary; just a yellow filter illuminated by the red and green primaries. Thats, like, total hokum.
If by represent, you mean encode, yes! However, they certainly cannot be displayed. Imaginary primaries, which really stands for physically non-realisable colours, cannot be used to build a physical display that encompasses the Spectral Locus, i.e. more than three are required.
Now the reality is that those highly saturated colours are not very common in natural surfaces, a natural reflectance gamut such as Pointer's Gamut or SOCS show that no chromaticities reach the Spectral Locus.
Colours outside Pointer's Gamut are typically created with highly saturated emitters such as LEDs or lasers. Because they are spectrally sharp, Observer Metamerism is noticeable and two persons might experience very different visual perception for the same stimulus.
Given we are trichromats I don't get why we would need a 4th band to cover more colors instead of continuing to expand the current 3 bands until the resulting color space encompasses our own.
Short answer: because the space of real colors is convex. It cannot be fully encompassed by any linear combination of three (or in fact, any finite number of) real primaries.
You can entirely encompass it using imaginary primaries, which is what some color spaces do (e.g. ProPhoto RGB), but it is physically impossible to manufacture a linear primary-based display which does. Adding extra primaries however does greatly help. DLP did this by having 6 (I think) primaries, but I don't think it was widely taken advantage of.
For a visual representation of the problem, look at the first diagram on https://en.wikipedia.org/wiki/Color_space and imagine how to encompass the colored region using a linear combination of three (or more) points within said region. (You can't.)
For the long answer, study up on human tristimulus response curves and how those interact to create the Planckian locus on CIE xy colorimetry diagrams.
