In general we leave the anchor secured in some fashion when running the boat as well as when trailering. I have never run over an anchor or accidently dropped it--but have heard of it happening. I agree that it is more likely to happen in a power boat. The CD 25 we own came with a chain stopper--and no windlass. I removed that--and we just use a line.
When we were long distance cruising we had a bail over the anchor, just to be sure it didn't jump the roller, if we stuffed the bow. On a passage we might either remove the anchors, and stow them aft, or in the bilge or lash them to the anchor roller/pulpit. Frankly, I don't trust windlass clutches all that much! Actually mega yachts do have remotely activated chain stoppers, so this alread exists.
Despite having sailed over 200,000 miles and gone over 25,000 miles in power boats, we have never come on a circumstance where we had to immediately launch an anchor. That doesn't mean that we didn't have emergencies--but that anchoring within seconds was not the solution or prevention. You can always go thru the foreward hatch if you have to pull a pin in seconds--and if you do that you can manually anchor if necessary. Remember the windlass is not designed to holld the load of the boat at anchor, nor to break the anchor out of the sea bed.