Why is it impractical?
A bike, like a motor vehiclle, displaying a number can be identified.
Any attempt to sell it without a log book will be very difficult.
Anyway, you say it is impractical, prove it, let us hear your reasons.
There are some crucial differences between bikes and motor vehicles:
No one under 16 can drive a motor vehicle on a public highway, all motor vehicle on the road have someone they can be legally registered to which makes it possible to make it compulsory for all motor vehicles to be registered. Bikes are ridden on the roads by minors, if they were exempted from the registration regulations then there would be unregistered bikes on the road. So it would not be enough to identify bikes without a number, only bikes without a number that are actually being ridden by an adult would a sign of an offense being committed.
There are very few retailers that sell cars sealed in factory boxes, bikes are frequently sold that way. Either the retailers would be forced to register every bike, which could significantly add to cost (for example a £10 registration fee and £5 to cover the cost of registering and fixing a plate would add 15% to the cost of a £100 bike), or the registration system would have to be easy for customers to register their own bikes.
There would have to be an easy way for, say, a 13 year old to register his bikes so that he can carry on riding them when he turns 14, or for someone to register a second hand bike that was previously used by a minor. Registering a bike built from new and used components would also have to be cheap and easy.
All in all the law would be impractical because to identify bikes breaching it the cyclist would have to be stopped, and if the bike didn't have a registration plate the rider's age would have to be determined. It would not be a simple case a spotting vehicles without plates.
Buying an unregistered bikes and then registering it yourself would be routine, so it would nowhere near as difficult to sell a bike without a log book as it would be to sell a motor vehicle without one.