The whitestra audio system has two audio amps: one for cabinet and one for backbox. First you need to rule out a bad speaker. You can hook a 9v battery to the speaker leads and verify you hear tone. Once you determine its the board, you should make sure the header pins are ok. And if so , you may need to swap the tda2030 that runs the backbox.

When you install the new speakers, make sure you don't drop under 4 ohms. This means standard car speakers for the backbox will need to be wired in series.

When I changed the cabinet speaker for STTNG I noticed the backbox speakers are 4 ohms each, and they are wired in parallel. That would result in 2 ohms in total. This was installed like that from the factory and more WMS games have it that way.

The fact that Williams sound engineers were idiots doesn't change the fact that the tda2030 amp is a 4 ohm stable amp only. It says so clearly in the part data sheet. I haven't looked at the schematic for the Williams DCS sound board but perhaps it has a resistor inline to prevent impedance dropping too low. Many 4 ohm speakers are actually closer to 6 ohms and often tweeters have such little draw that in practice they don't strain the load (Williams games have a mid range and a tweeter). It is wrong to wire two matching 4 ohm car speakers in parallel in a Sega game. Why would you want to put extra stress on a 20 year old sound board that's no longer available to replace?

You mean the 5 pin amp? Just order and solder in a new one

That looks pretty bad. So yea get a new one. http://www.marcospecialties.com/pinball-parts/TDA2030A

Make sure your speaker isn't shorted before testing the new one. Looks like something blew that up like a direct short.