Your cycling fallacy is…
“There shouldn't be any provision for cycling because people who cycle don’t pay for the roads”
More cycling benefits everybody in society, even those who never travel by that method. More people getting around by cycling means fewer cars in traffic jams, more space on public transport, less motor-caused pollution, and a healthier population in general.
It's also worth considering that most people who cycle do also use motor vehicles – very few people travel exclusively by one mode of transport – and are therefore also paying motoring taxes. Cycling also causes almost no damage to the roads, which therefore require fewer repairs.
In most countries, highway networks are built, repaired and maintained from general taxation, and the duties paid for fuel, vehicle taxes, etc., are used for a variety of purposes, like any other tax. Even if motoring taxes were hypothecated exclusively to 'pay for roads', that would still leave a huge financial black hole for the social and societal costs of driving, such as the harm caused by pollution and collisions.
In any case, we don't expect people who walk anywhere to pay a tax for walking along footways – so cycling should be no different, especially as more cycling benefits everyone in society. Similarly, should only people who travel at night be expected to pay for street lighting, or does it benefit us all?