I don’t think there’s a need to have a gun stored at your house for clay pigeon shooting. Store it at the range, under what I assume are more secure arrangements.
You’re right though, it would make me uneasy having a gun in the house rather than provide me with any sense of security. My understanding of gun security is that you probably won’t have the time to get it out and prepared in the event of a break in if it’s secured properly. (But happy to be proved wrong).
I think that's rather the point, having a gun for security is not an acceptable reason to own one. If you tried to get a license on those terms for a shotgun it would be denied.
Family members of mine own shotguns for clay pigeon shooting and even I think it would be safer for them to be stored at the range, albeit maybe not that practical because a lot of people shoot in different places regularly. The requirements for storage at home are quite strict though, e.g. ammunition has to be stored in a separate safe as I understand it. And every few years a police officer visits to inspect your house and assess any risks - If you're not in compliance you can lose your license easily.