TL;DR: Cheap powerbanks are overrated
I’ve bought a few powerbanks like this one:
These are the common problems:
1. Outdated and underrated SMPS
These power banks have to convert the battery’s voltage from 3.0V – 4.2V to the USB’s 5V. You can see the inductor with propably 3.3uH at the bottom left just above the USB-port.
This is physically not large enough for 2A * 5V = 10W.
Additionally, you can deduce from the winding that it has a large stray capacitance which forbids high frequencies. And a low frequency is common for old, low efficiency converters.
You could fix this with a chip inductor and an up to date converter IC in the MHz range – which costs money and some expertise.
2. Grossly overrated Lithium Battery
Chinese vendors “overrate” their batteries. The factor 3 is common.
Worse, those batteries deteriorate quickly. I’m not expecting more than 50 charge/discharge cycles from this one. The failure mode is an increased internal resistance which expresses itself in unreliable operation when the power draw is at maximum.
Not every manufacturer does this. F.e. my xiaomi powerbank truly has the rated capacity, charges with a high current at high efficiency, and additionally can be charged and discharged simultaneously.