I use a slightly modified rig that is easier to store. It consists of a bouy, a carabiner and a small piece of climbing kit.
The little bit of kit is an ascender but, although usually alloy and having moving parts, there is a version that isn't from Petzl. It consists of a chrome plated bent piece of steel with spikes.
I clip the krab to the bouy, put the rope around the rope and clip the krab through the ascender. Then use as normal like an alderney rig.
The anchor comes up and stays up suspended under the bouy when you stop and you simply pull the surface rope into the boat and lift the anchor in.
I also us the same system with a weighted grapnel hooked into wrecks although you need to take up the slack until taught before giving it welly to bend the hooks and release the grapnel. It you just bombed away when it caught it would probably do some damage!
This has worked for me for many years and once you start driving away the anchor lifts straight up so little chance of catching it on the sea bed!
As I said I lift grapnels straight out of wrecks using this so the anchor/grapnel is in mid water immediately as long as you take up the slack first.