But though we be the children of technology

Solution 1:

The 'but' at the start of a sentence shows a contrast / surprising disjunct with a statement in a previous sentence / main clause. I'll invent a suitable situation.

[B] Though John is smaller than average, he excelled at basketball.

One can't insert a 'but' into this sentence as a standalone sentence.

But now with a previous sentence (or main clause, if one adds a comma or dash):

[A] Taller people have obvious advantages in some sports. [B] But though John is smaller than average, he excelled at basketball.

The 'rule' you've been given as an aid, like many, has its limitations.