If the double wound transformer primary 'shorted' to the transformer frame/chassis then the chassis could be at the mains potential.
The transformer would most likely continue to work OK.
Remember, that these classes did not always exist.
Take a simple electric metal post lamp. It was probably never designed as class II. lamp cord goes through the threaded rod and in the bulb housing there is a layer of a cardboard like material between the lamp housing and where the wires connect.
But checking your link it has nothing to do with electrical appliance classes, which perhaps explains your confusion? - notice we're discussing class-II - NOT class-2.
Take a simple electric metal post lamp. It was probably never designed as class II. lamp cord goes through the threaded rod and in the bulb housing there is a layer of a cardboard like material between the lamp housing and where the wires connect.
1978 was long after double-insulated appliances were commonplace.
As for an antique electric lamp, if it's metal like that it's almost certainly class-I, even if such a thing didn't exist when it was made. One of my jobs is PAT testing, and I would test it (and fail it) as class-I - it simply needs rewiring to be safe, and to pass a PAT test. I do a lot of lamp testing (and rewiring) for a local antique shop - and you don't want to know the prices on the lamps!