Of course we will survive anything the climate throughs at us.
For one thing, it will occur gradually.
If we can survive in space we can survive on an icy planet.
You are correct, it is a crock. The last time the entire earth was under ice was 3 or 4 billion years ago, snowball earth, doesn't happen every 100,000 years.
You need to do more research. I recommend reading "Not by Fire, but by Ice" by Robert Felix. There are many more cycles documented by Felix. The article "cherry picked" the big items. The volcanic activity causes the oceans to literally boil into a "fish stew". The extra water vapor precipitates as snow. The snow and ice cover reflects sunlight causing temperatures to fall and persistance of the snow/ice on the ground.
Fresh water creatures near the equator/tropics survived the last ice age just fine. Woolly mammoths frolicked in the snow too long and were caught in snow too deep to escape. There are huge piles of woolly mammoth bones where they were trapped by the cold weather.
The effects observed are strictly consequences of the normal orbit of the earth and movement of our solar system through the galaxy. No magic. No politics. Simple physics.
BTW, there are regular layers of iridium that match these periodic events. The famous "KT" boundary is just one of them.
Technically, we are in an ice age. Ice ages are segmented into glacial and inter-glacial periods. Right now, we are approaching the tail end of an inter-glacial period. Lots of things survived ice ages, just in different regions and had to repopulate areas after the ice retreated.
It happened so rapidly that those older plants and animals were cryogenicly preserved. They revivified when the Proto-SUVs came out of hibernation, and caused Global Melting.
If you need proof, it is indelibly inscribed on their genome, and that is the reason they tend to sleep, or if more deeply affected, estivate, or even hibernate, when it gets cold.
That is even the reason YOU sleep at night better than during the daylight hours: It is the coolest part of the day. It also explains why unairconditioned humans sleep poorly in the summer months--it is too warm to sleep.
As for "no global mass extinctions", you are forgetting the Great Drowning that wiped out 97% of all ice worms. If you don't believe me, just look at most any active glacier, and you will see the remains of their abandoned tunnels.
The melt water flooding out of those tunnels washed the worms into the rising oceans, where they drowned and disappeared without a trace, either eaten by fish; buried in pelagic mud, or simply rotted. Only their tunnels are left to tell the story.