Aiwa HS-JX101

Fixed the broken flex cable with some l-shaped solid wire, and resoldered the switches connectors cable - neatly! A dab of hot glue (not shown) finished the repair, so the flex cable could have some... flex and not break

Colour me impressed with that flex job
Every time I tried soldering onto the flex cable I kept burning the amber plastic part of the cable.
You could write a whole guide on how to fix Aiwa's. I'd read all of it.
Kudos

@Command8 that's likely because you have a soldering iron that's too hot. It is critical for such delicate components to have a temperature that is, like Goldilocks says, "just right".

I use this Weller WESD51 station.
For soldering and desoldering this kind of flex cables I set the temperature at 320-330°C.
For thinner flex cables (e.g. LCD) it's around 250°C or even lower.
For desoldering components it's around 370°C, maybe as high as 400°C for old, oxidised solder.
For soldering new components, it's back to 330°C, though it varies a bit depending on the actual solder I use.
Whichever the temperature, I don't linger on a particular spot for too long. I'm done in 2-3 seconds. If it didn't come out right, I clean it and start over again.

These values and equipment should give you the right setup to not burn through the flex cable. Then it's time to sharpen those skills to make it look nice and work too.