Pretty much exactly correct. If you want to delay a signal, you have to store it for the entire length of the delay. That's impossible (or takes impractical physics) to do with purely analog electronics. All the solutions people have used for this in the past come with their own limitations; tape is unwieldy, noisy and relatively acoustically inflexible (haha). Classical bucket brigade chips are semi-analog, but are noisy, leaky, and have to be run at a very low sample rate to get a good amount of delay time. Tubes of mercury work well too, but you need to have a tube full of mercury in your basement (you get the picture). What we do, instead, is convert the analog signal to a bunch of digital samples and store them that way; our society has lots of great electronics for doing that bit. Then, a little bit later, we convert exactly the same samples (in exactly the same order) back into analog signals. The trick, though, is that we're modulating the sample rate so that sometimes those samples get played back at a different speed than they went in. They also get mixed and filtered entirely as analog signals, so there aren't any digital processing artifacts from those functions.
