3rd Gen Macbook pro randomly suspends

Your battery is dying. It's time to replace it.

You have very similar symptoms to the issue in this other post Fully charged MacBook Air turns off when power cord removed, but the major difference is that you have enough "juice" to allow macOS to suspend rather than just turn off all at once.

One of the biggest factors in batteries dying is it's age. This is a 2013 MacBook Pro, so it's got 6 years on it. The average lifepan of a battery is 3 to 5 years so (some last only 2 while some go 6 or 7), you're outside it's usable life.

Battery utilities

Coconut Battery, et. al. are neat little tools, however, they can only give a point-in-time calculation of your battery's heath. It does it by measuring the current (volts and amps) and the current draw. What it can't do is measure the charge like a fuel gauge in your car.

This is why you're seeing the (confusing and conflicting) "battery's ok but system sleeps when not plugged in" symptom.