How would the world be affected if precious metals became far more plentiful?

There would be winners and losers.

The winners: Scientists that use gold foil and wires in satellites. Electronics producers use precious metals in electronics. Car manufacturers use platinum in catalytic converters. Basically anyone who needs to buy precious metals to make a product.

The losers: Jewelers wouldn't be able to make much of a profit. People who own precious metals will see their value decline. Countries on the gold standard would have to break from it. Gold would no longer be used in international trade. All the gold a fort Knox would be almost worthless.

Money itself has value because other people accept it as payment. This shouldn't change. You would still be able to buy things with silver coins as long as other people accept them as payment.

It would probably benefit the economy because precious metals have many uses, but there would be profound social changes.

There would be an impact, but not all that great; and it would depend on which metals and how cheap. All in all, whenever any material becomes cheaper, people use it more.

Silver is a better conductor than copper, so if it were as cheap as copper, people would use it for electric wires.

Gold does not corrode easily so people used it to protect electrical contacts, etc. If it became very cheap, it would be used for plating larger areas, etc.

Platinum, palladium, rhodium, and osmium are very good catalysts for many processes. They would be more widely used as such, making a goodly number of goods significantly cheaper.

They and gold are denser than lead and non-toxic, so if they were cheap enough they can be used in everything from shotgun shells and fishing weights to gym weights and tire weights.

Note that when you say "industrial metal level" prices, that can mean anything from low carbon steel through copper to tungsten, cobalt, and titanium, that is, more than a factor of 200 to 1.

steel is about $500/ton

aluminum is about $3000/ton

copper is about $8000/ton

nickel is about $18,000/ton

tin is about $21,000

cobalt is about $100,000/ton

http://www.kitcometals.com/

http://www.metalprices.com/