There is no benefit in doing so unless you are developing a low power portable system where the case and mainboard are integrated to the point of marriage.
The issue with heatsinking being the case is you don't just need surface area, you need conduction rate which means the thickness of the metal. It is much more cost effective to move the heat away from the component with the majority of the mass right at the heat source then fins radiating outward from that.
So the question you asked is invalid. It's not that a case can't be a heatsink, but rather that it is just some novelty idea to try to do it with no practical benefit for a typical PC. If you were talking about some special avionics or military module that needs to be sealed then it will be in a thick metal case designed specifically for that as I mentioned initially, completely integrated not a standard ATX consumer grade etc.
I could run a very low rpm large fan to cool it that would be completely inaudible.
That has nothing to do with whether the case or a separate heatsink is used. You can literally slap a big ole audio amp heatsink on your case panel if you are just too bored to use proven working solutions to cooling, or remote mount a water pump where you can't hear it, to a large enough radiator that you don't even need a fan. However, I have built many systems with heatpipe sinks, including overclocked, where I never hear the 120mm heatsink fans. It's usually going to be a case wall mounted fan or PSU or especially video card fan that makes the most noise that escapes the case.
There are all kinds of wacky ways to make life hard on oneself instead of just using what everyone else does. Besides, if you make it too quiet then you'll be cursing at crickets for disturbing that.