Fiat coins materials aren't changed in value to match their face value. They're changed to lower production cost and achieve a particular quality of production.
That isn't entirely correct. The mint changes coin composition to reduce the intrinsic value of a coin to be less than face value. It is classic debasement. Gold was removed from coins in 1933, silver in 1965, and most of the copper in 1983. In all cases the coin composition was changed because the intrinsic value was greater than the face value.
Paper and ink do have a marginal intrinsic value. It just isn't that much. In areas with rampant counterfitting, the value of a printed bill settles to the market value of the paper and ink.
Because the mint has a budget and the cost of metals impacts that budget. It is not due to considerations about giving the coin any particular intrinsic value.
Your statement and mine are in alignment. Metal content is changed to reduce the cost to make a coin which results in an intrinsic value lower than face value. The reason that the mint changes coin composition is that the existing composition has an intrinsic value higher than face value. The Mint counts the difference between melt and face value as revenue. If a coin is intrinsically worth more than its face value it is a loss to the Mint. So, they change the composition to be less than face value so that coin makes them revenue again.
Paper and ink do have a marginal intrinsic value. It just isn't that much. In areas with rampant counterfitting, the value of a printed bill settles to the market value of the paper and ink.