Up thread there's also a citation of the Betamax case, which says that it's okay for people to record shows so they can watch them later, and it's okay to sell video recording devices.
This is different from Bittorent, which automatically distribute any content you download. With this too, when you are downloading, you are also distributing, which is a much clearer case of infringement in most jurisdictions.
What private copy wanted to cover is more like "I bought a physical CD and I make a cassette copy because my car doesn't have a CD player" or because I want to listen to a mix of my favorite music (so you paid a tax on cassettes and later on CD-R and CD-RW media). This was later extended to "I copy all my collection of music to a hard drive and stop flipping CDs in and out of the player" (so you pay a tax on hard disk drives).
YouTube's paid offering in some countries includes this feature on Android and iOS, but for computer users there's no built-in way to do it. Wouldn't it fall within the spirit of the private copying rule for a subscriber to YouTube's paid service to use youtube-dl in this way?
You are wrongly assuming that a copyright violation has taken place. If a country's laws say it's legal to make a copy under specific circumstances, it's not a copyright violation to make a copy under those circumstances.
Of course YouTube has other videos that are under different licenses so it isn't clear that this is the only use case of the software.