One of the windings has a capacitor between
the 'red' contacts, with one of the leads also
soldered to the black contact which seems to
have an effect of shorting a part of the winding.

You may be misreading which wires are parts of which windings. It's
not too unusual for two independent windings to be connected to each
other in funky ways to choose 120VAC vs 240VAC inputs, for example.

Big leap time: plug it in, see if you get AC out of the transformer and
DC volts across the filter caps.

Tim.