Right now, I say 8GB VRAM is enough for 4K gaming. Once game engines start making full use of next gen consoles hardware, then it will be a different story, but by that time, the next gen GPU will be out. Obviously more is better for future proofing, but again that depends if you plan to hold on to a GPU for a couple of generations, or upgrade at every generation.
You will have to consider, the more VRAM you have, the more VRAM caching will take place. This is not the amount the game needs to run, just the system caching memory if it is available.
Whilst I'm awaiting Zen3 and either a 6900xt or 3080, I have a makeup PC with a 5600xt 6GB, powering both a 4K monitor and my LG 38GL950 3840x1600.
In games I have tested maxed out at these resolutions I have not hit more than 5.8GB VRAM,
This includes SOTTR, BFV, Hitman 2, RDR2 to name a few. So take that as you will, but proof we are not limited, even at 6GB in certain games that I've tested.
From experience of hitting VRAM limits in the past, you would pretty much end up with a slide show, not a few frames dropping off.
To see other games at 4K maxed out on a 3070 where it chugs along just fine, there is lots of videos on YouTube with comparisons of other cards side by side in real world scenarios.
That said, many people I know who game at 4K don't usually run AA, so VRAM usage will be less. Me personally, I hate TAA, makes the image very blurry, where I like high clarity.
https://youtu.be/lXIkgk6sOHw