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"no-one has answered where the horizontal reaction comes from"
yes they did, pushing against friction of the pivot and possible air resistance
"Everyone agrees that with a rigid rod it's possible to generate horizontal movement. A bicycle operates in a similar way in principle."
a bicycle would be like pushing off the ground on a swing, you are fixed to the pivot in all axis and pushing the pedal/swing. Not touching ground on a swing you are only fixed in the vertical axis.
"The question where people still differ is what happens on an idealized swing in a vacuum with a frictionless rope of zero flexural rigidity."
My instinct says you can't start the pendulum.
" how do the kids move their centre of mass with no reaction force to start with?"
they don't