Without knowing your situation I can only advise following some basic troubleshooting technique:

Check the wall voltage with a meter.

Follow the wire from the wall outlet to the service outlet in the front of the game. Check the voltage at the thermistor etc.

Check the voltage at the on/off switch.

Check the fuse in the service box with a meter or tester.

Follow the wire from the service box or on/off switch the the PC. Find the ends, check the voltage.

Try to at least for now to see if there is actual voltage at the internal power supply board via the wires you traced.

Get back with the results and pics of your work so far.

Booting the PC is not a simple on and off proceedure.

Its an ATX system.

If power is available at the power supply and everything seems in order, then the power switch pins on the mainboard have to be briefly jumpered to start the current flow. This can be done manually or thru a relay etc.

If the ATX power connector is connected and power is available,
Im curious if the PC will start by jumpering the switch pins on the mainboard.

The PC isnt too complicated, if the ATX power connector and the 4 pin CPU connector has power available, it should boot, unless there is a mainboard fault. It seems that you have a replacement mainboard and it didnt fix the problem?

As was mentioned before, check the connections and check the power everywhere.

On a hunch I would suspect a power supply failure ( main fuse, broken wire, thermistor, bad power board? etc) of some sort before I supected a mainboard failure.

You stated that the game booted with the old board, but there were errors.

Does the game boot with the old power board installed again?
I would re-install the old board and see if the games starts, even just temporary for a minute or so, regardless of errors.

If the game starts with the old board then the fault is in the new board or the installation procedure.

Sometimes when working backwards you will find the loose connector or a pin of a molex that pushed out of its housing or a broken wire.

It is possible for a power supply to kill a mainboard but, its very rare.

Generally if the power supply fails then no power is allowed into the mainboard at all and its protected.

In the event of a catastrophic short (melt down, lightening strike etc.) then the mainboard will fail but you will see blackening of the ATX connector on the mainboard. And it may trip the wall breaker.

The majority (but not all) of the mainboard components are rated for 250V but low current ratings, so yeah a direct short can be devastating. But again its pretty rare.

The coin cell battery, whether or not its installed, will not keep the PC from booting, its purpose is to keep your CMOS settings intact so you dont have to type in the time, date (and specific settings perhaps), each time you start up the PC.

In the OP's case the pc isnt booting yet. All of the other errors can be looked at after it boots.

Thats what I was thinking as a test...
Or maybe put the old board back in as a test to see if it boots.

The OP says its dead.

Sounds like no power or service box fuse.

Testing the power at the service outlet would uncover that fault.