I don't expect the replacement op-amps to make much of a difference. A replacement bridge rectifier of this size costs sub-$0.50, so it would be an easy thing to try. There's not a whole lot that affects the voltage from the transformer: input voltage, winding ratio, core, wire gauge and load. Since this is a shunt regulator, the current draw is going to be fairly constant at the voltage across R132 divided by 330 ohm for VR105 to maintain regulation.
You don't need a dual rail power supply. Just apply +24 VDC to the AC terminals, measure, and then apply -24VDC. Essentially a diode test on steroids, although you might as well measure the voltage on the DC side while you're at it. The scope screenshot suggests maybe conduction between the two AC terminals at +24 VDC. Or an open in the diodes conducting at the negative part of the cycle.