This higher-resolution plot on Wikipedia does seem to show an Eemian peak temperature of approximately +4 degrees C. The dataset is the same (EPICA Dome C) so I think that the much longer timescale of the first plot I linked to simply averages out relatively short-term peaks like the Eemian.
With that said, I agree that the modern pace of climate change is much faster than these historical cases, and I share your concern that changing the climate so quickly might trigger something that slow changes didn’t. I’m just saying that most of the Earth isn’t going to become a desert.