With out getting into childish arguments about this or that aircraft being “copies” of another (they are almost never such) I would note that the IL-76 entered service approx 8-9 years after the C-141 and 4 years after the C-5 Galaxy.
So “the Wests” (US) airlifters were always as advanced if not more so than the Soviets equivalents. This is not a criticism of the IL-76 or the later AN-124 (probably better than the Galaxy but at least a decade later so it should be), both well proven designs.
Given that at the time other western countries couldn’t afford (to buy and operate) or decided not to prioritise their own C-141 or C-5 equivalents (the Belfast and its travails and its 10 year RAF service being the exception that rather proves the rule) then seems unlikely any non-US IL-76 equivalent would have emerged in anything like the same timescale (the programme that became the Atlas is considerably later). The international sales of the C-17 are also a much later post-Cold War phenomenon (and a non-US aircraft would not have been realistic given the volume of non-US orders).