With load sharing, the USB V+ source automatically takes over powering the load circuit when it is plugged in, and the battery charging proceeds to completion independently. This is necessary to prevent continuous charging of the battery, which could lead to smoke and flames.
The module in your picture doesn't appear to have load sharing parts, and I haven't seen a module that has charging, protection, and load sharing. But load sharing requires just three parts - a P-channel mosfet, a Shottky diode, and a resistor.
Here are two links that tell you all about it. I think they use different charger ICs, but the principle is the same. You just have to be careful to orient the mosfet so the body diode is pointing the right way.
thanks, I was not aware that the part I found had no load sharing or "power-path".
So, in a nutshell, one has to solder one's on circuit, as I understand the Zak site. I, obviously wrongly, thought that - because there are so very may Li-Ion products out there these days that one can use while plugged in to charge - I could buy such a "charge-while-use" PCB off the shelf.