Quoted from barakandl:

I suppose you could put a screw terminal at every single transformer wire connection. I wouldn't recommend it, but you can do whatever you like. Another brand does this. I would be worried the corroded wires going high resistance burning up. A few connections are going to be high current, but you can find appropriately rated terminals.
The issue with the transformer wires is they are often corroded. You strip back the wire many inches and it is still all corroded and will not want to accept solder. You can fray out the wires and try and to abrasively knock off some of the corrosion. Re tin the wire. Then solder it to the board. Make sure to heat the circuit board hole and wire at the same time. Once hot enough the solder should just flow nicely. If you can't get the solder to flow, try a bigger iron tip and increase the temperature. This circuit board is made with double thick copper and the traces are all very wide which act as a heat sink sucking the heat off your iron tip.
Someone makes a new transformer to rect wiring harness, sorry I forget who. If the transformer tap wires are really nasty, consider getting new ones. Should make it easier to solder.