Yeah that is why Silver was a medium of exchange, everyone mined it and everyone used it. They didn't really NEED silver, per se, they just accepted it because that is how currency works everywhere at this point. Almost everywhere in the world bases its currency on something rare in order to make counterfeiting difficult. China isn't special for not having enough silver to meet demand, that's true of almost every country in the world.
As for taxes being paid in silver it is technically true but not really. "paid in silver" is just a crude way of saying "pay your taxes in cash." All money was made from silver in China and it was on the silver standard, just like everyone else, so even the occasional copper, gold, or alloyed coin just represents a certain value of silver. Hell, that's how all bimetallic currencies operate. The British Pound for example, despite being made of gold, represented a value of one tower pound of Silver because that was the standard all currency was pegged to.
The reason China became a black hole for silver was because it's the only they'd accept historically. They didn't want European trade goods, there just wasn't a demand for it. That's part of why Opium filled that niche, it was one of the first goods in demand in China that they couldn't make themselves and it was a hell of a whole lot cheaper for Europeans to get than silver AND you could put more of it in a ship since pound for pound refined Opium was more valuable than silver.
Chinese silver scarcity was also due to a hoarding culture prevalent in the country. The wealthy would just lock up their silver and not spend it, so the silver didn't circulate. Once silver arrived in the country it'd just go into some rich guy's vault. Rampant corruption of this sort was a serious problem in the later periods of most Chinese dynasties, a period which coincided with European trade activity in the Orient during the 18th and 19th centuries. The most egregious and famous example was an official named Heshen who served in the later 18th century for I think a bit over 20 years. He had in his hoard silver and gold and other goods valued at around 15 years of the revenue of the Qing government. It just sat there, in a hoard, not doing anything.