I have no opinion on the quality of the U10 and U1 chips shown below. Most Nanos don’t have your problem. There may just be a stray wire strand accidental jumper on the circuit board so you should look at it closely with a magnifying glass. You may also want to measure R27 to be sure it is 3300-ohms.
If it were me, I would cut the foil trace between BAT+ and U1-pin-3, install the battery and see if it runs down. If it does run down, then the problem is probably U-10 if no stray wire strands were found.
If it doesn’t run down, then resolder the broken trace and replace U-1.
If you look at V1 schematic, the part inside the dotted box does not exist according to ESP. You could maybe adapt that circuit if you can’t get the special chips. The disadvantage of the V1 circuit is that you can overcharge and damage the battery if you leave the live USB cable attached more than a couple of hours. That could also present a fire hazard.
