I had a similar issue on the exact same computer type - really frustrating to always have to take out the battery and put it back in to turn it on. Turns out it was the replacement keyboard I bought because the caps lock key on the original one was loose.
If it is indeed the keyboard, then don’t do what I did and purchase an extremely cheap one with a backlight that broke after installation. Doubt it’s a battery or power related issue since Apple devices usually automatically power up when you plug in the charger.