Do cyclists pay road tax?
No. Though cycling organisations campaigned for decent roads to be built when there were about 3 cars in the entire country.
cars pay tax for several reasons, the wear they cause to the road (a cycles is negligible), they pollute and they are inherently dangerous (though i wouldn't go as far as making them all drive behind a man with a red flag again)
You appear to suffer the fallacy that paying car tax entitles you to the road, when the majority of roads in the country were not designed with cars in mind, they are for all forms of transport.