A solar system is formed out of a mix of gas and tiny nanoscopic dust (and ice) grains. Those dust grains contain the whole variety of elements that exist in nature, from hydrogen to uranium, including things like gold. As those grains stick together and form into balls of rock and ice there is a certain size range where the energy released by impacts starts becoming enough to start melting the material of the grains. As you progress through the ladder of accretion of larger and larger objects (from gravel to rocks to boulders to mountain sized asteroids to moon and planet sized objects) eventually the heat released from these impacts becomes enough that it can keep a whole object molten for an extended period of time, not just an isolated melting but a full molten asteroid or "planetesimal".

When this happens the grab bag of materials in the dust grains that made up all of the component parts of the object become liquid and start to separate based on mutual solubility. Just like oil and water don't mix, different kinds of molten elements and minerals either mix or don't mix. In broad strokes you can categorized elements and minerals into the group that is more soluble along with molten iron and nickel (the siderophilic elements and minerals) and the group that is more soluble along with molten silicate rocks (the lithophile elements and minerals). When a large planetary body exists in this molten state for an extended period (perhaps as short as hours or days even) it separates into layers of materials based on mutual solubility and density, with the denser iron and siderophilic materials sinking down into the core and the lighter silicate rock materials floating to form a mantle and crust.

Many precious metals including gold, platinum, palladium, iridium, etc. are siderophiles, and preferentially separate out into the core of a planetary body. Earth contains an enormous amount of gold and other precious metals effectively locked away within the core, on the crust gold is much rarer due to this differentiation process which has occurred. During the formation of the solar system some planetesimal bodies became large enough to become differentiated but then cooled to become solid, and some of them were disrupted by impacts and their pieces ended up not becoming parts of planets or moons. Today there are multiple types of asteroids. Many are made of very primordial material (chondrites) and were likely never part of a larger body and never underwent differentiation. Some are the debris of larger bodies which became differentiated and then were smashed into pieces by impact events, leaving behind floating chunks of stony crust or metal rich cores.

The metallic asteroids are fragments of the cores of larger differentiated bodies, and they represent a peak into the sorts of collections of materials that on Earth are mostly abundant within the core. Psyche does not have the density to be purely a chunk of iron-nickel core material, it likely is some mixture of metallic and stony materials. However, the metallic materials will include mostly iron, nickel, and cobalt.

(Here's the tl,dr part):

The precious metals within 16 Psyche exist at trace levels, with higher concentrations within the metallic portions of the asteroid. For gold, platinum, iridium, palladium, etc. these levels would typically be somewhere in the 10s of parts per million range. They do not exist as native metals or in the form of flakes, grains, veins, or nuggets. Those require hydrothermal processes which do not occur on asteroids or asteroid parent bodies. This means that in terms of precious metal content even a high metal content asteroid represents merely an ore body that would need to be processed in order to produce bulk quantities of those precious metals. It would take processing tens to hundreds of tonnes of asteroid material just to produce a singular kilogram of gold or other precious metals, for example.