Earlier quoted context omitted.
I was wondering how Docker-in-Docker works, but I couldn't find it dockermented anywhere. If it's using the host's Docker daemon, why do you need to mount the docker socket?
Docker-in-Docker (DinD) doesn't piggy back on the host's Docker daemon, but instead runs a stripped-down Docker daemon inside of the container. The major downside is that I/O is quite slow, since you're going through two virtualization layers (the DinD one, plus the host Docker daemon).
There is, effectively, no "virtualization" layer here. There are some things that if needed can cause overhead... such as the bridge networking (really shouldn't be a bottleneck for majority of people), and the CoW filesystem... which docker won't be (or shouldn't be) running on top of since, for example, overlayfs on top of overlayfs is not supported.
There is also nothing stripped down about the daemon inside of the container.