Climate change skeptics often imply that since cold and warm periods existed long before humans, ...

The fact that past climate changes were not caused by people does not prove that people cannot change climate, much the same way as the fact that forest fires happened long before humans does not prove that a human cannot start a forest fire. In the ice cores, the temperature during glacial cycles correlates with CO2 and methane concentrations, and with the timescales of orbital (Milankowitch) cycles. This might mean that small warming on account of the Milankovitch cycle could have been amplified by increasing concentrations of GHGs. So if small "natural" increases in CO2 and methane could have added to the warming, then the much higher "human-made" increases in these gases we have now would have even more pronounced effects on the climate in the near future.