Dropped phone doesn't turn on unless connected to USB

Hello,

Just dropped my phone on the floor, it went black and wasn't turning on. After I connected it to USB, Google logo appeared on the screen, blinked 3 times and it booted normally. The second I disconnect USB, it goes black and unresponsive again.

I assume there's no power output from the battery?
When the phone is on (with USB), I can see the battery charge and charging icon, so it's not completely disconnected.

1. Should I try to check the battery connection?
2. If connection looks good, does it make sense to try to replace the battery?