There are a lot of different HP gaming 15 models e.g cx- range, dk- range ec-range models and I'm not sure if they all use the same DC-In jack connector if that's what the problem is. e.g. it could be a resistive connection in the jack which may prevent the correct voltage from appearing. The USB only needs 5V but the motherboard needs more otherwise the adapter wouldn't be rated at 19.5V.
The exact model number should be on the info label on the bottom.
Also not having the cmos battery plugged in will cause problems because the cmos battery maintains the BIOS settings and helps prevent them from becoming corrupted which can prevent a normal start.
Not having the main battery and using the AC adapter only should mean that the BIOS is reset every time the AC adapter was disconnected but it takes a little while to reset unless a forced reset is used so maybe it wasn't being reset properly every time.
Check what the voltage is on the cmos battery. If it is <2.5V DC replace it.
If OK before reconnecting the cmos battery, disconnect the adapter and press and hold the Power on button for a full 30 seconds and then release it. Reconnect the cmos battery, reconnect the adapter and check if it now turns on.
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