So that ROM is an I2C ROM, the MC has some I2C physical compatibility on the SPI port so those 2 wires could be routing over to pins 17 and 18. That ROM is likely storing the settings of the welder, how the MC/Welder would respond to a new blanked ROM is anyone's guess tbh. In fact how the MC/welder would respond to a faulty ROM would be guess work as well, one would like to think the firmware on the MC handles it by at least putting out a fault code on the display but ....
That said one of those lines is a clock and could be used as a diagnostic to see if the MC has some sort of alive pulse.. so if you got a clock signal on Scl (of the I2C) that would suggest MC is powering up and initializing.
Do you have an Oscilloscope/logic analyser btw? Also importantly some of the safety training/self thought/experience required when working with mains equipment?
Just to add to a previous post regarding the LEDs/7-segment displays, i made the point that the LEDs and 7-segment displays are driven from the MC and therefore the issue points strongly to the MC UNLESS there was a common-supply. Looking a bit further and trying to work out how the LEDs and 7-Segment displays are driven that point about a COMMON-SUPPLY is something that I'd be inclined to check.
1. As far as i can tell the LEDs/7-segment display are actually part of the same drive circuit, on the 7-Segment displays you you have a total of 3 x 8 =
24 leds effectively. If you look around the board you can count 20 status/configuration Leds
populated and 4
un-populated =
24 (or visa-versa 3 more 7-segment displays with period)
2. There is no way the MC is driving all of these separately, there wouldn't be enough pins on the MC to do this. The MC is probably using 8 pins to drive all LEDs and the 3 7-segment displays. (you can look up circuits on how this is done if your not familiar). This brings back the point about the common-supply to these LEDs/7-segment displays, if that supply is dead than you get nothing on the welder front panel.
3. One notable place where this common-supply could be is on the ULN2803A transistor arrays (Pin 10 on the datasheet). I'd certainly probe pin 10 on these ICs to be sure. There does appear to be zener-diodes located on the PCB very close to these transistor arrays, these could be providing the common-supply voltage, its not impossible that one of these is dead.
Without schematic and the board at hand I am assuming the transistor arrays have anything to do with the LEDs/7-segment display, i am really just basing it on their geographical location on the board. Its tricky to trace PCB tracks with pictures only

Regardless I think you have gone the right route for now in contacting a supplier and hopefully getting the support needed to get the welder working.
I would be interested if you got any response from that supplier, were you able to get a decent price for a replacement or even a schematic?