The two most important elements of a modern Justice League origin seem to be: (1) a threat big enough to unite the heroes; and (2) a reason for why the Justice League operates independently of the government.

If DC were to write a modern origin for the Justice League (in a setting as realistic as the MCU), how would they come together, if the government didn't recruit them to be a team? And why wouldn't they work alongside the government or the military when facing a particularly dangerous threat?

In this scenario, there were no superheroes operating or working together before the Justice League founders.