The answer to this is that the 1 cent coin (penny) being the only copper coin is a recent thing.

The original US coinage, authorised by the Coinage Act of 1792, included a copper 1/2 cent coin as well as the cent. Here's an example from 1793. These were produced and circulated into the 1850s.

There was no nickel originally. The original 5c coin was a silver half disme, issued in 1792. This makes sense given that the purchasing power of 5c in the late 18th century US was fairly significant - maybe US$10-20 in modern day money. US coinage was not fiat money until 1873, in practice. A copper coin which actually coinage 5c worth of copper would have been enormous; Great Britain issues a copper twopence coin in 1797, which would have had roughly the same value...and it weighed 60g, with a diameter of 41mm. The technology to mechanically manufacture these very large, thick coins was in its infancy, and they were quickly abandoned in the UK due to unpopularity.

Other metals than copper or silver would not have been obvious to the early US mint. Billion coinage (debased silver) did exist in Sardinia during this time period, as well as some German states, but tended to wear very badly. Experiments with tin coinage in 17th century Britain had been disastrous due to quick development of corrosion. Nickel had only been isolated in 1751 and was not used in coinage anywhere until Switzerland started issuing cupronickel coins in 1850.

During the Civil War there was widespread hoarding of precious metal coinage, which commanded a premium over Union/Confederate banknotes and various types of local currency. This continued in the aftermath of the war. The economic disruption resulted in the issuing of various non-precious minor coins from the US mint, including a copper 2 cent coin, from 1864 to 1872, and a nickel 3 cent coin, which circulated from 1865 to 1889, and replaced an earlier silver 3 cent coin. The five cent nickel was introduced in 1866, and the silver 5c, which was not being produced in enormous quantities, was discontinued in 1873. The US nickel, ignoring changes in design, is basically the same coin today which was issued in 1866. The modern US cent, which is smaller than the original issues, has been minted from 1857.

As to why the other coins aren't issued in different coloured alloys, this is largely because the 10c and 25c were minted in silver until 1965, and therefore they chose to use cupronickel and later cupronickel cladded copper, in order to maintain a similar appearance while removing the precious metal.

Source: History Of The United States Mint and Its Coinage, Lange, Mead and Bowers, 2005.