I have downloaded the zip file and seen all the pictures. There is no galvanic isolation between triacs and control circuit so it is unsafe to touch any part of the control circuit when power plug is in the socket.
Turn off the machine and take out the power plug from the wall socket and do the checks below.
1. Inside the the violet jack (close to the optocoupler and triacs) there are spring type metallic contacts. It is possible that one of them is damaged when the pcb track is burnt. So make sure using ohm-meter that the violet jack gets proper electrical connection.
2. On the circuit board, check the resistance of triac gate series resistors R41 and R37 (these resistors are connected at pin 15 and pin 14 of ic KID65004). The DMM's readings (100 ohms) should be nearly equal. You don't need to desolder the resistors. This check will make confirm that none of these resistors is open circuited.
3. On the circuit board, check the resistance of gate-MT1 capacitors C30 and C31 (these capacitors are connected between each triac's gate and MT1). The DMM's readings should be nearly equal. You don't need to desolder the capacitors. This check will make confirm that none of these capacitors is short circuited.
4. Pin 2 and 3 of ic KID65004 are the control inputs for the triacs. It seems that the pcb traces between these control inputs and the main microcontroller 87CK14 are damaged. See the attached picture below.
If the problem is not solved yet then you need to measure voltage in running machine at various pins (pin 15,14,2,3 respect to pin 8) of ic KID65004. Depending on the readings we can decide the ic KID65004 is faulty or not. If it is found faulty then we can replace it by either original type or an alternative ic ULN2004. To avoid risk during measurements you can solder one end of a wire at a pin and take out the other end from machine. But first, need to know the results of the above 4 points to be checked.