It's not that stupid.

What would happen if there was a big private blockchain "secretly" double spending?

Say it currently had 48% and was a few blocks behind the original blockchain -- and everybody knows it.

What's to stop you mining on the short chain but spending on the long chain?

Now let's make a Big Assumption: people bandwagon on this new chain if it's at 48% in order to selfishly double spend when it finally takes over the old chain.

If that is the case, then surely it is rational for me to do the same thing but when it's at 47%
and if that is the case, then surely it is rational for me to do the same thing but when it's at 46%
...
etc

They're stating that the initial threshold is actually very low. They're stating that rational malicious entities mine on the short chain even if the short chain has, say, 5% of the computational power.

I don't know how they take multiple private chains into the analysis or what constant they're using for that Big Assumption.

It does sound a bit dodgy to me; I don't think there's reason to believe this will happen any time soon. But the paper isn't completely without thought..