It's trivial to make an image that you can see the 8-bit quantization in. Just make 256 vertical gray bars, equally spaced. In much of the image (usually the darker parts, but depends somewhat on monitor calibration) you will see the border between two values. Note that this is every single possible grey value in 8-bit RGB, so it is a proper un-dithered rendering of a gradient, and the borders you are seeing are definitely quantization artifacts.
Now, if you apply proper dithering, particularly on a high-dpi screen, you can make it look a lot better, possibly even undetectable.
And this is just with the dynamic range that is comfortable on a computer monitor! If we really want to represent vibrant images, then the glare of the sun off of shiny objects should be much brighter than the "paper white" background used in e.g. your word processor. It should be intuitive that the brightest highlight in a real-world image ought to be too uncomfortable to fill your entire monitor with...
Lastly, there's a question what and how the image gets to the browser. Sure for the images on the landing page for $LARGE_CORPORATION you have a graphic designer make everything look great, and since most people still have 8-bit displays, you'll want something that looks good on one. But what about e.g. a site for uploading photos? Are you going to quantize all of their photos before displaying them (possibly to someone with an HDR display)?