It all comes down to supply and demand.
The supply side is pretty straightforward. Video games in general, Nintendo games, and especially Gamecube and its games were popular at a much different (less mainstream) level than they are currently. Animal Crossing was a top 10 Gamecube game in terms of sales, yet sold only 2.7 million copies.
Take a look at the top selling Gamecube games to get an idea of how many copies best sellers sold in that era.
And then compare that to this list for Switch. New Horizons has sold 40 million + copies to date. That's a 15x difference in sales for AC on Gamecube vs. Switch. Taking into account that most copies of Animal Crossing for Gamecube were sold 20 years ago means that some number of copies are broken, missing, thrown away when people moved, long forgotten/buried in a box in public storage, or gone for any number of other reasons. The number of remaining and available copies only goes down over time. And of those remaining copies, some number of those belong to people who would never consider selling their copy of the game (like yours truly). So as time goes on there's less and less supply available.
If no one cared about Animal Crossing for Gamecube, that limited supply wouldn't matter too much. There are 20 year old collectibles you can buy online for relatively low prices. But when demand goes up, especially significantly, for an older, out of print item, that limited supply leads to higher prices. Assuming there are now 40+ million Animal Crossing fans worldwide based on New Horizon sales, if even a small subset of the 2023 ACNH fans want to play the original, it's easy to see why the price of Animal Crossing for Gamecube (and likewise for other Gamecube games) are at the levels they currently are.