1 amp of what? See, herein lies the problem. You don't specify, and perhaps don't understand, the specific metric. Did you measure charge current, or discharge current? Continuous or peak? More importantly, note that the current could also be limited on the side of the power supply (e.g. a usb port, but also any smps will have overcurrent protection) or even on the battery side (if it is a pack that has integrated overcurrent protection circuitry).
Likely none of the products you evaluated was "fake", but you simply were a little nonchalant in evaluating them.
In any case, for your custom pcb, I think you should have a decent idea now of how to approach it. I don't see a reason not to use this particular IC, combined with the apparently popular dual mosfet or a different mosfet of your choice.