How does that work with the "time shifting" laws that allowed VCRs to be used? Isn't this basically the same thing?

One problem is that you're not only time shifting but space shifting to digital media as well. And the laws as written are very specific as to what is permitted.

Example: In the US, the AHRA that makes personal space shifting digital audio legal (relevant to the submission) is fairly limited in the media that can be used for copying. Though I agree that they're being used for basically the same thing, computer hard drives aren't covered.

A relevant quote from Wikipedia [1] that sums up why I think this act falls short of its purpose: "In each case, the principal distinction between what is and is not covered by the AHRA is determined by whether or not the device is marketed or designed (or in the case of media, commonly used by consumers) to make audio recordings, not the device's capabilities."

[1] https://en.wikipedia.org/wiki/Audio_Home_Recording_Act#Digit...

I don't get the distinction. Why is saving a stream to an mp3 space shifting but recording a broadcast to a vhs not?

Time shifting means copying for the purpose of listening to the recording at a better time, whereas space shifting is relevant here because the intended purpose is to allow the audio to be played in a different environment (i.e. not the YouTube player or app).

The more important thing is that space shifting to different media (digital, personal computer hard drive in this case) is legally treated differently. Though, to be honest, I'm not convinced that any personal recording protections would or should apply to the YouTube situation. It's just one way that the two differ.