Forcing an ATX supply to turn on by bridging the green wire to GND is a very bad idea for a pc setup. PC mobo's have a voltage rail supervising circuit that checks all the voltages on the mobo, not only those coming from the supply but also the voltages generated on the motherboard used to power the cpu internal Logic. When such a voltage is out of range, the motherboard will turn off the supply using the green power on wire. So, if you short that to create a permanent on, the system can't turn it off and you can fry your cpu or ram modules if a voltage is 2 high.
Motherboards have a pair of jumper pins on the mobo that are provided for a "power on" pushbutton. That's the recommended way to try to turn on the system. Further, they might have a setting in the bios that tells what to do in case of a power failure. If that's set to power on, the mobo should turn on the power supply as soon as the power is restored. Obvious, you need a good bios battery or it won't work. So, just buy a cheap meter that can do volts, amps and ohms to check the voltages. Measure the CR2032 coin cell battery voltage. (with the power turned off) and see if it still gives you 3V. If it's only 2.8, replace that. Find the jumper pins that are used to connect the power button. Usually, they are grouped with also some for a reset button, a power led and hdd led. If you are lucky, it's printed on the pcb what they are used 4. Temporary short the power button jumper pins and see if the cpu blower does an attempt to rotate. (Even if it's only for a fraction of a second.) You should be able to find an ATX supply for 50 - 60 bucks. Maybe computer shops will even loan you one for testing, or you could bring the TTX to them and ask them if they could test it. Sometimes they have an ATX tester that loads the supply and shows it's status with some led's.