The cheapest way is allot of running water, then IPA wash, then remove with a fibreglass pen. Repeat rinsing and washing as required when using fibreglass pen. Once you get rid of all the corrosion replace missing traces. And through hole plating.
If you have to remove allot of contaminated PCB you will need to replace it with high temp epoxy. 3M sells EpoxyPatch for allot of money but it is basically pigmented high temp epoxy.
Applica Traces and through hole grommets can be overly expensive. For non professional jobs I have been experimenting with alternatives. My best hope is using copper tape.
Once affixed and prepared tin the new traces. If you do not have tinning solution then tin with solder. But the temp is critical. Too hot and the traces will lift. Too low and you do not get bonding.
Next over coat the traces with PCB lacquer, then clean the pads. Or cover with high temp epoxy. If you don't have a hot air tool and you need to use a heat gun you may need to remove more components.
Once dry install components, clean then lacquer again if you used lacquer previously.
This is the basics of board repair. There are many more steps but these are the ones I foresee you will require.
Testing continuity test will only test for conduction it will not test it ability to handle the specified current load.
In reality the only real way to test a component is a curve tracer. But a component tester is the most common accepted method. IE an LCR meter. Continuity across anything but a resistor is no guaranty that the component is good. With out a component tester or curve tracer you will need to find the part through symptoms and processes. Such as using lower AC input, change the bridge to a resistive load, applying DC to the AC input to name a few.
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