There are several reasons why we don't make coins out of precious metals, and also a few why we should. The wastefulness of resources is relatively low down the list.

The point of a commodity-backed currency is to provide stability in the money supply. Provided the commodity is widely regarded as valuable, but isn't likely to change in raw value in the long term, having x amount of money represent y amount of that commodity means that your money is a safe store of wealth. If you want to earn lots of money now and spend it in the future, you can be confident that inflation isn't going to devalue your savings in the interim.

This is one of the reasons the price of gold is negatively correlated with equity performance. In Bear markets, people buy gold because they believe it's less likely to lose value than holding your wealth in stock or currency, and generally they're correct to hold this belief. In part, this then becomes a self-fulfilling trend, as gold itself becomes a good investment at the start of market decline. Partly because of this, you'll find armchair investors are especially keen on the Gold Standard, because that's what they think money is for.

There are problems with this, though, namely that inflation isn't just a monetary phenomenon, and currency isn't just a store of wealth. Money is also a medium of transaction. There has has to be enough money, in sensible units, to represent the value of all the goods and services that are traded with it, so as the economy grows, there has to be a proportional increase in the money supply, otherwise your currency will appreciate to unwieldy levels and you'll suffer from effective deflation.

Gold is a bad candidate for fulfilling this function of money. There's a finite amount of it. We've already mined most of what's easily obtainable, and eventually any given good or service will be represented by an increasingly small quantity of gold. Any changes to the gold supply are going to be sudden shocks to the economy (say I irradiate a whole bunch of it like in Goldfinger, we discover a massive new supply of it, or a huge gold asteroid drifts into Earth's orbit). Also, part of the usefulness of gold as a basis for currency was that it wasn't used for a whole lot else. As various industrual uses were found for it, that impacted both those industries and the currency. If we found out tomorrow that eating gold cured cancer, increased penis/breast size, and gave people superpowers, that would be bad news for a gold-backed currency.

The purpose of the bitcoin mining process is to provide a stable supply of bitcoin. Since each block adjusts the difficulty of producing the next based on the efficacy of mining currently taking place, the production of bitcoin should be continual but stable, (at least until we mine them all).

I am still thinking about the ramifications of bitcoin, and what properties it has as a currency, but the method of bitcoin production is one of the things that makes it novel and special as far as currencies go. It can function as both a store of wealth and a medium of transaction without those functions necessarily working against each other as they do with traditional commodity-backed currencies.