Thankfully, this is a topic that's well documented/explored here:
www.downtowndougbrown.com
Your Mac II is starting, so you know the battery part of the circuit is OK - it wouldn't start without that being OK. I think you need to poke around the bits in the 'Power On' section of the circuit:
First off:
The 7V rail also goes to the emitter of transistor Q3, which is a PNP transistor. We’ll see in a minute, but when the power button is not being pressed, the transistor is turned off, so it shouldn’t be drawing any current.
The interesting bit here is Q3 - if that transistor is off, the power supply is off. And that transistor is supposed to be off until the power switch is pressed.
According to that page:
Since 7V is going into the left side of R14, there is no current between the emitter and base of the transistor. The transistor is turned off.
Because the transistor is turned off, the power supply control pin (J18, pin 15) is left floating (remember, the +5V output is not supplying 5V yet, since the power supply is off).
Thus, the power supply stays off.
Hopefully that will give you some ideas of other bits to check.
Before repairing it behaved correctly.
You mentioned replacing a broken power switch before, what made you think it was originally broken? What other work did you do? (I can see the batteries have been replaced), was there anything else?
A photo of the back of the board may help myself or others identify a potential problem area.
I hope that helps. It's a lot to get your head around, but approach it methodically from the power switch through to each step.