Why "silver-tongued" for someone who is convincing?

It does not refer to the value but to the "sound" produced by silver objects:

• Of voices, words, etc., from 1520s in reference to the metal's pleasing resonance; silver-tongued is from 1590s.

Etymonline

Silver tongue is a phrase used in the Bible. Proverbs, 10:20, King James Version

The tongue of the just is as choice silver: the heart of the wicked is little worth.

The OP asks: but would gold not have been more apt, as it is worth more?

Golden-tongued has been used, the most famous examples that of a 4th century saint and a 19th century U.S. politician.

Golden tongued, or, more exactly, golden-mouthed was appended to the name of St. John, Archbishop of Constantinople. See Wikipedia, John Chrysostom

John Chrysostom ..... c. 349 – 407,... Archbishop of Constantinople, was an important Early Church Father. He is known for his preaching and public speaking, his denunciation of abuse of authority .. by both ecclesiastical and political leaders, the Divine Liturgy of St. John Chrysostom, and his ascetic sensibilities. The epithet Χρυσόστομος (Chrysostomos, anglicized as Chrysostom) means "golden-mouthed" in Greek and denotes his celebrated eloquence. (Emphasis added.)

See also William Jennings Bryan: Golden-Tongued Orator by Robert A. Allen, Peggy Trabalka (Illustrator). (WJB, 1860 to 1925.)

Describes the life of the lawyer, orator, and politician who ran unsuccessfully for the Presidency three times.

As for silver- vs gold-tongued: @ Mick says in his comment

The finest musical instruments were originally made of silver.

But that pushes the OP's question to: why were the finest musical instruments made of silver and not gold? It may be simply because gold is so much more expensive than silver, and thus gold instruments were exceedingly rare.

@JOSH said in his answer that silver has a pleasing resonance, but gold musical instruments also have a fine tone, as the following reference on gold flutes discusses.

A flute can be made of even 24K gold. From The Flute Center of New York

Most flutes made of gold or platinum have a silver mechanism, but for those willing to go the extra mile, we have flutes with gold mechanism as well. The added weight of a gold mechanism will certainly affect a flute's resistance, its weight, and its price! While 14K gold is most common, many flute makes use other alloys, like 9K, 10K, 18K, 19.5K or even 24K because of the unique tonal characteristics of each alloy.

The ultimate is a platinum flute (See, e.g., Altus Platinum flute) , but platinum was not known in ancient times, and, because of its very high melting point, probably could not have been worked. so platinum-tongued never became established.

If the OP wants to pursue the dominance of silver musical instruments over gold musical intruments, https://musicfans.stackexchange.com/ or https://music.stackexchange.com/ may be a better place to ask.