Would anyone care to speculate why a three cent piece was needed in 1837?

Your idea a of three cent multiple chart is a good one. The Spanish system, with the pieces of eight, had far more influence in The United States than people realize. For example, the fare on the Roxbury coach to and from Boston was 1 reale. In 1837 the Spanish dollar was legal tender and would remain so for another 20 years.

The three cent piece would have fit in with the reale, 12 and 1/2 cents, and the half reale, 6 and 1/4 cents. I suppose that the fractions of a cent were ignored.

 

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When the first half dismes were coined in 1792 for Thomas Jefferson, you can see it in his daybook where he recorded every expense.

The evening after he picked the coins up and started out for Montecello he tipped the porter at the inn 35c. You can't make up that sum with 1/4, 1/2, 1 and 2 reales can you????

 

I had read this before, contains some different information than that article
==> The 1837 Feuchtwanger Cent - Quality Collectible Coins
https://qualitycollectiblecoins.com/blog/the-1837-feuchtwanger-cent/

 

The 1792 half disme was the introduction of the decimal system of coinage in the United States. The reale was part of that system only to the extent that the Coinage Act of 1792 called for a quarter dollar which was 2 reales instead of a 20 cent piece which would have been more in keeping with base 10.

The trouble was the first United States Mint was too small to supply enough coins to the country. Its problems were made worse by the fact that the gold coins were too heavy relative to the silver coins, and the whole idea of a bi-metallic monetary system was unworkable. It took years for the American officials, as well as officials in other countries, to figure that out.