So, I'll have to insist that they comb them out
Some knots are NOT comb-able no matter how hard you insist.

Clipping through the knot (as shown in the picture) is absolutely correct. The brush SMALL sections at a time (think "inch by inch will be a cinch, yard by yard will be hard"). Use a 'picking' motion versus trying long smooth brush strokes. Start with a wire slicker brush and work your way to a comb.