An AC component present on any DC bus on a complex digital device such as a pc will cause immeasurable and erratic behavior. As far as I know, the typical life expectancy of electrolytic capacitors is somewhere around 2000 hours +/- or 5.5 years.
The internal power supply within the device will contain electrolytic capacitors used to filter hum or AC ripple. These capacitors will decay over time especially if the environment is warm. The dielectric layer between the plates (rolled foil) dries out over time (faster with heat) and the capacitance value decays. Eventually the capacitor is not able to adequately filter out the AC component and any resultant AC component will as previously mentioned, screw up digital circuitry.
That's just one theory that could apply.
Here's another potential idea that could support the pc not starting for a period of time after the battery is removed.
A semi-conductor device within the battery charging circuitry has a failing component such as a voltage regulator.
Connecting the battery with an external power-pack enables the internal charger, the faulty voltage regulator suffers a thermal shutdown and kills the main or secondary B+ line, some LEDs might work but the pc will never boot or POST.
Wait 5 minutes for the faulty regulator to cool off, remove the battery so the charger circuit is no longer active and presto, the PC boots.
Now that was an overly simplistic theoretical explanation for a relatively complex device. Consider there are most likely 2, 3 or 4 different voltage regulators all regulating a different voltage bus, (-5vdc, +5vdc etc) for different devices within this PC. A failing semi-conductor anywhere in this device could cause erratic behavior as you've presented.
Given your description, my money is a fault with the internal charging circuitry. If the internal power supply is built with micro-circuitry like most devices today, it's most likely not worth your grief to repair. I've had some limited success with these type repairs but they're often a huge PITA!
Move your data, pop the top off and use a laser sighted thermometer to measure for hot spots. I'm retired and occasionally I might embark on one of these adventures, maybe. My Dell laptop was a huge PITA to disassemble/re-assemble. I'd use it as a paper-weight before pasting the cpu again.