You should also note that the circuit simulator you're using, and LTspice, don't always give results that can be believed.That actually helps me understand what I'm doing. I'm going to look into common-mode with regards to amplification circuits and learn more.
Looking at the schematic for LM741, you can see why an input voltage of a few millivolts won't work.
From Texas Instruments:

You have 5 transistor junctions that need to be properly biased for the input stage to function.
LM358 from Texas Instruments:

It was designed to operate from a single supply. Since the input stage is PNP, ground is allowed. You still need to stay a couple volts away from the positive supply because you have 2 BE junctions and a current source that need to be biased properly.
When operated from dual supplies, a resistor is needed between the output and the negative supply to minimize crossover distortion.
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