That probably is a bad battery then. Li-ion batteries can catch fire or explode if they're overcharged, or over-discharged and then recharged. So there's a lot of circuitry added to prevent these things from happening. They also have a very flat discharge voltage curve - their voltage does not change much from a full charge to what's considered fully discharged.
When you first turn on a computer, the act of powering up circuits and devices like the screen draws more power than keeping those things powered on under normal operation. When you draw a high load off a battery, it depresses the battery's voltage. The voltage is probably dropping below the battery protection circuitry's threshold, causing it to immediately shut your Chromebook off because it thinks your battery has been over-discharged. When you have the AC adapter plugged in, it is providing the power to cover this initial spike, the battery voltage remains stable, and there's no shut-off. Once the Chromebook is running, its power draw would then be low enough that you can unplug the AC adapter, and the battery voltage isn't depressed enough to cross this threshold. So it remains on.
I would try searching online for guides and videos on how to change the battery on your model Chromebook. It's not trivially simple like adding RAM to a laptop, but on most models it is one of the simpler repairs you can do.
If replacing the battery fixes the problem, you can dispose of the old battery at Staples, Home Depot, or Lowes - they have recycling bins for Li-ion batteries. It's still possible for the battery to catch fire if it's punctured or damaged, and it contains some toxic chemicals, so you don't want to just throw it in the trash.