Why is sync so important when making a bootable linux usb stick?

The dd does not bypass the kernel disk caches when it writes to a device, so some part of data may be not written yet to the USB stick upon dd completion. If you unplug your USB stick at that moment, the content on the USB stick would be inconsistent. Thus, your system could even fail to boot from this USB stick.

Sync flushes any still-in-cache data to the device.

Instead of invoking sync you could use fdatasync dd's conversion option:

fdatasync

physically write output file data before finishing

In your case, the command would be:

tar -xzOf archlinux-2016-09-03-dual.iso | \
dd of=/dev/disk2 bs=4M status=progress conv=fdatasync

The conv=fdatasync makes dd effectively call fdatasync() system call at the end of transfer just before dd exits (I checked this with dd's sources).

This confirms that dd would not bypass nor flush the caches unless explicitly instructed to do so.