This is a total waste of time and money for Nintendo. They can't possibly shut down every ROM site. Nintendo also cannot completely get rid of ROMs that were acquired or shared in violation of some changing set of laws. What Nintendo should do is focus on providing a superior experience.
Nintendo's current emulation of old titles is frustrating. Some titles are only available for 3DS, some specifically for the New 3DS and not the original 3DS, some only for Wii U, and some only for Switch. There's also games that were only provided on Wii and never made available on newer platforms. Worse, buying a classic game from Nintendo locks it to the platform it was bought on, so if you bought an old game on Wii U then you'd be unable to play it on Switch without re-buying it. Nintendo promotes and distributes their back catalog in such a way that downloading ROMs from a website and playing via a bulky PC is the most convenient approach.
Nintendo should evolve and improve the process of enjoying their back catalog by making as many games as possible available for Switch. The Virtual Console should have no arbitrary limitation of what you can and cannot play. Give people the ability to play the old games, tie purchases to their account rather than their hardware, charge a fair price, and provide a straightforward experience that just works. This his how Nintendo could beat the ROM sites. No, I'm not paying $10 to play Zelda ALTTP on the New 3DS again (I have an old 3DS), but I would be willing to pay $3 to $5 to play it on Switch.