I`ve not had a TR7 stalk apart but it all looks fairly similar to others such as Vauxhall Magnum and the like (albeit the stalks are opposite hand).
If a TR7 stalk follows the same system then the push button you have has a contact ring that when depressed touches the two copper legs, one is connected to the aluminium stalk arm (its alloy under the black finish), the other to a wire that runs down the center of the stalk tube. At the other end of the stalk a wire, in the main switch body!- (usually purple and black) is staked to the alloy tube outer and this and the center wire runs to the plug/socket with the others. In my experience the wire staked to the alloy outer can fracture at the staking point with time with usage and then no circuit can me made to sound the horn.
Does this seem to be like yours, can anybody confirm if this description makes sense for a TR7 stalk??