As an electrical engineer I can tell you that those cheap sources of batteries are exactly the problem. The reason that some of these batteries are so cheap is that they leave out protection circuits that are intended to shut things down in case of a fault so that no overheating (fires) can occur. Consumers can't tell the difference between batteries with and without the safety circuits. They buy on price alone.
I seriously doubt if any of the "review" sites even care about this aspect of safe product design. Only reputable manufacturers such as brand name PC product vendors care because their brand name is likely to be damaged if they sell products that catch on fire.