Oh they are. But heat pipes/vapor chambers have them beat. So they are only used where electrical insulation (pure diamond sheet) or mechanical strength (cheap diamond grains sorted into a densely packed single layer of diamonds embedded into a copper matrix; this fits nicely for silver sinter bonding dies to due to the low thermal expansion of diamond) is needed from the heatsink.

Riffing on this, the amazing thing about diamond is not merely that it's a better conductor of heat, but that it's about 2.5 times higher than copper.

The only thing that beats it is single crystal boron arsenide which has some obvious downsides, being of course much more expensive and involving arsenic, which is considered a bad thing.

And with that, diamond is destined to be even cheaper than silicon, I think. Much more available and easier to purify, the hard part is rendering it into a crystal which is relatively solved.

No for PC heatsink, but in 3d printing heat is really important for the nozzel. And a company makes them with Polycrystaline Diamond, which is harder than normal diamond. The company needed it themselves as the carbon fibre filament they were using kept wearing out heads. They though "Why don't we make our own heads".

https://www.youtube.com/watch?v=96eFnTescoY

It’s still prohibitively expensive for such applications.