This is a fairly common problem we've seen with the power supplies in Mac II-series machines. See my recent discussion in this thread. In my personal testing, I've isolated to problem to not include capacitors, fuses, batteries, motherboard problems, etc. Rather, the relay units on the power supplies themselves often fail to function properly with age, becoming either sticky or refusing to make contact altogether. In your case, that's very likely what's causing the machine not to shut itself down automatically: the relay fails to disconnect even after the power off instruction is issued.

I have found, in my extensive work on some IIsis, that often the connector from the PSU to the logic board gets corroded and prevents the power supply from making proper contact. Plugging and unplugging it REPEATEDLY will often solve the problem, but in particularly tricky cases, a straightened paper clip and perhaps some rubbing alcohol is useful to scrape out the inside of the connector holes. This solved an issue with a couple of power supplies that appeared completely "dead" and/or refused to soft power off.

However, your startup issue sounds more like the issues commonly noted with the IIcx/IIci/Q700 series PSUs, as described here. There have been mixed reports of fixes, but I am unfamiliar with the II, IIx, IIfx power supply personally. Perhaps someone else here has more direct, hands-on experience?

Also, equill points us to this very helpful page on which you can find schematics of some of the II-series.