Forget ebikes, buy a 50cc scooter.
why is everyone throwing in the towel on this one so easily ?

.. if I buy a scooter it would be at least 125-150cc with a capacious lockable under-seat storage compartment

Seriously though, I appreciate you cannot make a thing perform like something it inherently isn't.
However .... I'm hoping you can get a lot more for less or the same money with a self-assembly approach than a ready-made. At the moment it seems that's the case anyway. Priorities and objectives will narrow down further as more information is processed and more research completed.
The objective is trying to avoid petrol use and seeing whether you can still feel like you've a great quality underlying bicycle capable of performing with or without power assist, along with power delivery which helps at those times you really need it most (or to speed up a challenging route so as not to leave you debilitated) rather than just at times when it would be 'nice' but you could personally probably manage without. Plus having something which does not unduly enslave you to a charging point every few miles (so to speak) in the way early electric cars did. There just aren't enough charging stations yet (although some great efforts at trying to change that are clearly underway and I fully support any moves to develop that network).
Much has been posted on the forum by people who have and clearly still are achieving great results with their converted bicycles, using the technology and products out there to get better and better results for the same outlay - without getting petrol involved and without spending thousands of pounds. It's made me much more determined to bottom it all out for myself as much as I can and then put the work and effort in to get the best result achievable with the benefit of the very valuable experience and input members on here, as far as they are willing or available to help, once I've sorted my own head out about priorities and options.
Hopefully in time my own experiences if they lead to a build might add to the pool of others who might encourage people to try it. All the direct responses in this thread have opened up my thinking enormously and made me question a lot of the assumptions I've been reaching reading around other threads, which is a very helpful thing as all planning has to start somewhere and then be properly channeled.
Would prefer to side-step the continuous output / wattage complication (until a workable specification affordably matching or getting as close to aspirations as possible is identified). Max power output when you need it seems not as closely linked to motor ratings as I first assumed - and there do seem to be several workarounds for upping that to reduce necessary continuous power output for a given performance requirement. Which side of the 'continuous power output motor line' I end up sitting on for a given application remains to be decided for certain, more likely driven by available kit choice, price and compatibility than anything else.
The registration vs no registration debate is not really relevant to the underlying goal - no commercial interests to consider, political points to be seen to be making or 'sides' to sit on. It's all about the bike ! Pretty sure there will still be plenty of punters wanting ready-made bikes and willing to pay, i.e those opting to convert will have little or no real impact on ready-made sales.
The really big pressure point of range vs battery and motor cost/weight is reflected in the price and availability of ready-made solutions in multiples.
The other pressure point for me personally is "top-end" riding capability and component quality of the unpowered bike to make it a truly versatile quality end product for a very low price. It's not a big deal if you're riding 6-8 miles there and the same back some time later. But it really matters when you go out riding for 20-30 miles only partly assisted, or choose to ride entirely unassisted. Probably something I need to talk to some people in a regular cycling club about rather than on here though. Even reaching that conclusion is a helpful and constructive result.
Component quality in eBikes only seems to hit higher end when you hit the thousands price bracket. It seems (and maybe I am wrong) that almost all the focus is on the motor/battery with the bicycle really playing 2nd fiddle. That's understandable commercially but it doesn't have to be that way if you are assembling and not buying something off the shelf. A better compromise seems well within the grasp. It's at least worth fully exploring - imho.
Sorry to rabbit on. I blame it on the onset of the impending cold snap or something
