Bad battery? Or motherboard damage

Firstly, thank you to everyone that least looked at this.

my iPhone 5s has weird symptoms that could be from a damaged battery or motherboard damage

heres the problem, when the phone dies, plug it in displays Apple logo for few seconds than fades to just a lit black screen.

IF I take the phone and put it into dfu mode and restore it through iTunes the phone will do its thing then bootup correctly no Apple logo disappearing

but… when the phone dies it does the same with the logo, I’ve got a good feeling it’s the battery but decided to get an answer from people who know more than me