Iphone 5s Only starts when connected to wall, dies when unplugged.

WHAT IVE DONE SO FAR (You might understand better what happend if you read this)

So im trying to fix this Iphone 5s that didnt want to boot up (Applelogo for like 0.5 secound, black screen for like 5 minutes, flashes Applelogo again for ca 0.5 secound, ect ect) I guessed the battery is dead, but when i plugged it in without the battery (Wich from what i know should make it start up without it, only using the power from the wall) It did the same thing. I tried connecting it to my pc to try to reset it, since i guessed it was just corrupted. It didnt start. i think my usb ports are to weak. So i was just messing around with buttons while connected to a wall, and then i showed low battery. It crashed, and i tried again. After a few secound of a black screen (Backlight still on) i tried unplugging it and it stayed on. i rushed to my computer and plugged it in. it seems like i got it in some kind of Recovery mode (I could not see the itunes logo tho) i reset the phone, it seemed to work. when itunes was done reseting the phone (not recovering an old backup since i didnt have one, starting fresh basicly) i got into the setup (I had 100% on the phone, so the battery must have charged while i was letting it sit in the charger for about an hour before) , and i was thinking that i would work. then it happend. "Insert SIM card". as soon as i popped out the sim tray, the phone died, just like when i cut the power before. Mind you i was VERY carefull since i was afraid that it would die again if it got hit or something. After that, i could not get it working again. doesnt even boot at all when connected to pc. Now tho, when i plug it into the wall, it shows apple logo for like 5-8 secound, then a black screen for about the same time and then apple logo again. Forever. And if i unplugg it it dies.

Facts:

Ive got the iphone to boot, even if it died during the Insert SIM card step in setup (Maybe ios wasnt ready to boot normaly)

The battery isnt completely dead, since it has shown signs of atleast a little bit of life

I CAN get into recovery mode, but only when plugged into a wall, and i cant unplugg it to then plug into my computer, since it dies instantly.

I have tried using a USB 3.0 self powerd usb hub, still doesnt boot the iphone

So,

Do i NEED a battery, or will i end up paying for nothing? i HAVE another working iphone 5s, but i would prefer not to open it unless it is the known fix

Other things to note that might help:

I dont know if it is an original battery or not, but i cant see an apple logo on it anywhere and there is no thing to pull to get it out. the phone is worn, the sceen was somewhat loose and the screen is a little cracked, tho this should not be the problem

nothing looks broken or busted inside.

Thank you!