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Well, they're obviously not alone in this. Calls for registration of cyclists are made all the time. One shouldn't be complacent about it, as you never know what might happen, but it's exceedingly unlikely that this will ever be taken up (so convenient for people who want to use it as an excuse to kick something into the long grass). The DfT has long been quite clear that it would require considerable changes to primary legislation (as bicycles are judged to be carriages as per the 1835 Highways Act, you have a right to cycle, whereas for a motorised vehicle you need to acquire a licence--they're basically calling for the right to cycle to be removed, represented by 'licence plates' (although most people calling for that probably don't realise what they stand for and what changes they would require)) and would in any case be impossible to implement and enforce.
Needless to say, it is not something that only wealthy people call for, but a large part of the population regardless of income. It's a very interesting topic of how people feel about modes of transport. Fundamentally, people complain about their lack of freedom (i.e., the lack of a right to drive) as compared to the freedom that people have to cycle (because they have a right to). This is perceived as unjust by many, and while people of course exaggerate the perceived problems that cycle users supposedly cause in order to advance their argument, it is certainly not a complaint to just dismiss, but one to take seriously and discuss properly.
Fundamentally, it is clear that if there was a right to drive, this would cause very considerable problems, e.g. it might no longer be possible to attach conditions to people's purchase of a vehicle licence, or to require them to pass a driving test. I suppose these could be got around somehow (by introducing some other kind of mechanism), but the whole licensing régime is the main control mechanism to try to ensure at leats some minimal standards, and as we know, these are low enough as it is. Simply, it makes no sense to grant people a right to drive, as the checks and balances we have would still be necessary.
Conversely, that people have a right to cycle is, of course, not without problems (in the way that nothing is). The main problem that exercises opponents of cycling is, of course, the perceived lack of accountability (myths include 'paying no road tax' (Vehicle Excise Duty), not being insured (most cyclists are, either because they are also drivers or because they are members of a cycling organisation that includes third party insurance in their membership package), and legal differences, such as the applicability of speed limits (they don't apply yo cyclists), etc.
Calls to treat cyclists the same are therefore understandable in a way, and there are some things, like speed limits, where because there were very few limits under 30mph it was unnecessary to include cyclists in them, but as many riders can easily exceed 20mph, with the recent proliferation of them, the situation is now different (and that's not even to start on the whole can of worms of e-bikes). However, it is also clear that you need to treat different modes of transport differently (as they are different, natch), and most of the things usually demanded as part of a regulatory response to perceived problems with cycling are impracticable.
tl;dr--this one will run and run.
The residents of Kensington and Chelsea are so anti cycling they have mobilised and blocked almost every improvement that tfl has suggested.
Even putting Kensington palace gardens on a quietways map. Which would have made no intrinsic difference to anyone on a road that is closed to non resident vehicles.
Being well connected and organised really pays off..Even if that means you lie about being the aunt of cycling fatality in the area.
And now they expect registration plates!
https://road.cc/content/news/268357-no-safe-cycle-routes-until-riders-are-licensed-insists-holland-park-residents