That trace doesn't look like it connects vertically, it looks like it goes to the large square pad to the lower left of the IC Which means that the +/- IN voltages only go to the IC, and the battery leads are connected to the two lower left pins. C1 is a bypass cap for the IC. R1 and R2 are in parallel, not sure what they're for but must be some kind of feedback to the IC. Seeing as how the IN + voltage only goes to the IC it can interupt the current to the battery/circuit. Seems like a very basic charging circuit, I'm sure there are dedicated lithium charge ICs in that kind of form factor. If you're so concerned ask the maker to identify the IC.