Is my battery at fault?

I've inherited an iPhone 6 which a friend has recently upgraded from due to "battery problems", as I was told, and I'm trying to get it back into a usable state. When not connected to power, the device remains powered off completely. When plugged into power, the device seems stuck in a boot loop. It powers on, light grey background with Apple logo for about 15 seconds, then screen switches off for a couple of seconds, then back to the first screen.

I can put it into standard Recovery mode, but attempting to Restore & Update results in an error 4013. Putting it into DFU mode and trying the same gives an error 4005 (eventually, takes a lot longer). Tried on both Mac and Windows machines, a couple of different cables, just in case, but identical results. Device has no sign of any physical damage.

Now my question is: Does this describe a situation where the battery has (effectively) died, and what are the odds that replacing the battery will give me a fully functional device?