"An actual map may have billions of countries and need not even contain “a blob of one colour, wrapped in three others”."
In fact every map of interest will contain a blob wrapped in at least three others.
But they are right that the problem is showing that the theorem holds for maps with billions of regions, or more. Decisions you make early in the process of colouring the map can turn out to be wrong, but only much later as you are trying to colour the last few.
In short, you are saying "I can't see how it can go wrong", but that's not enough ... there are things that might happen that you weren't able to imagine, and that's part of why proofs like this are hard.
Also, the theorem is true, and that makes it hard to explain why it might go wrong. Because we now know that it doesn't.
The grand-parent comment also says:
"... the Jordan curve theorem ... is true on a sphere ..."
The stronger form is not true on a sphere. The stronger form says that every topological sphere in 3-space has an interior that's retractable to a point and an exterior that's retractable to a shell. But the Alexander Horned Sphere[0] is a counter example.
In short, weird things you didn't suspect can happen, so proofs are sometimes harder than you think, even for "obvious" things.
