When was it serviced, recently or in the 90s?

Yes, please post some decent pictures of the boards and check for the voltages that drocelot suggested.

Those look great!

Measure the voltages on the test points on the CPU board. Need to see if the power supply voltages are making it reliably to the CPU board.

Remove the screws on the driver board (lower board in your first picture). Carefully pull the 2 boards apart and press them back together. The long white translucent connector between the CPU and driver boards is the "40 pin connector", notorious for causing issues. Pulling it apart and putting it back together cleans the contact points and may be the cause of the problem.

Are the contact points in the battery holder rusted or corroded? They look rusted in the picture but it is hard to tell.

Yep, that’s right. The tiny metal studs are the test points. You can put the negative meter lead on the ground braid.

The game should boot to audits without batteries. Static numbers in the displays.

You can try and do the cheat boot, just in case the problem is with your displays. With the coin door open, turn the game on. Then very fast, turn the power switch off and back on. Attract mode should start (playfield inserts flashing). You may have to try it several times to get it to work. Based on your original description (both LEDs on), I don't think this will work.

With the game on, press the diagnostic button on the CPU board and tell us what the LEDs do.

You need to figure out why the voltage at the test points is so low. 4.7v is much too low to boot reliably. I think you are on the right track with reflowing the solder to the header pins. Make sure you do it on all the boards (PS, sound, driver, cpu)