So what exactly is the question? Can you turn the computer on without pressing the power button?
If so, the answer is both yes and no depending on how much semantics you want to get into. All you're really doing by pressing the power button is completing a simple circuit that bridges two pins on the motherboard somewhere. So, with the aid of a trusty flathead screwdriver you could manually short those two pins. That's where the semantics come in, because be it a press of a button or a flathead screwdriver, you're effectively doing the exact same thing, and if those pins on the motherboard ever are damaged in some way, you're really in trouble.
However, given the age of the unit and the symptoms, you may well have a bad caps (google it) issue.
Have Compaq Presario SR-1000 running great for over four years, do not turn it off, just into sleep mode when I am not using it. Recently it turned off occasionally when I hit the sleep button. I cured this by adjusting the screensaver power saver settings which seemed to vary capriciously. My next move to advanced settings proved costly. I was asked should the sleep button cause the computer to sleep--- well I guess that makes sense--- and then the killer, should the start/stop button stop the computer. I suppose you all who are reading this as well as I are wondering if the engineer who wrote this instruction or question is still employed ???

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