How well do different metals (specifically copper and aluminium) conduct heat to air?

But the stuff has to conduct heat through itself in order to conduct it to the air.

Utter crap. The rate of heat transfer to the air is utterly dominated by the conductivity of the air at the surface and the radiativity of the surface. And the area of the surface

Heat transfer does not have a 'speed'

Yes, but that's practically a short circuit with any metal heat sink

The picture in my mind is someone replacing the lead out wires to a 1K resistor with solid silver ones to reduce it's resistance.

Technically it will, but by so little its practically meaningless

It could be the color of the metal or how rough the serface is.

Forget air for the moment. Take metal A and metal B. Touch them together (very well, finely machined). What is the conductivity at that surface? The average of the two metals?

Why don't they make the surface of heatsinks really rough? As in like orange peel, more surface area?

The rate of heat conduction between two metals in contact which have different conductivities is determined by the thermal contact conductance coefficient. This coefficient indicates the thermal conductivity or ability to conduct heat between two bodies in contact. The thermal contact conductance coefficient is dependent on the surface roughness, surface cleanliness, and the nature of the materials in contact¹.

Source: Conversation with Bing, 20/05/2023 (1) Thermal contact conductance - Wikipedia.

The side of the heatsink which is in contact with the air is usually designed to have fins or other structures that increase the surface area. The increased surface area allows for more efficient heat transfer from the heatsink to the air. However, making the surface too rough can create air pockets between the heatsink and the air, which can reduce the heat transfer rate.

Indeed. Most heat is lost by a heatsink though conduction to the air which then convects it away. The smoother it is, the better the airflow