It is not that simple as you showed earlier in your post. One reason the Wandle Path doesn't work is because of the entry points. People will be prone to judge the safety of a route by the most dangerous part. If the danger is above their threshold they simply will not cycle. The safe bits don't count because they are - err, safe.
In London if you cycle more than a mile or two that is going to include a gyratory. These are a real challenge to the most experienced cyclist. A complete blocker for the novice.
Hence I do get irritated when LCC, Sustrans and my local council spend all their time and our money fiddling about with 'segregated' stuff in places where it is easy to do and merely convey the novices to the real danger points which are ignored just because they are difficult (and expensive) to sort.
If LCC named and shamed the worst ten gyratories in London and vowed to stretch their council across them until they were fixed - I would be very happy, Novices would be happy and many more would take to the road. We can the argue about whether we should join up the 'smoothed' gyratories with segregated paths later. But that's a very second order consideration in changing modal rates.
That's something TfL & LCC have been looking at as part of the junction review process - and some of them are indeed being fixed. Vauxhall, Elephant, Old Street, Aldgate all likely to get somewhat better over the next few years (jury still very much out as to how much, mind you), and I've heard they're looking at some or all of the Wandsworth system as well. Bricklayers already much improved (well, the on-road layout is still as shite as ever, but they've created huge, wide shared pavements on all sides & Toucans on the crossing arms - so less-confident cyclists don't have to tackle it at all). Elephant has the cycle bypasses a couple of hundred metres down the road.. they're far from perfect, but much better than the roundabout itself.
Those big systems cost massive money to fix & don't tend to be tackled in isolation, so, as best I can tell, LCC have to wait until the council / TfL / property developers etc. decide the time has come for major change, and then lobby hard to make those changes as cycle friendly as possible..
From personal experience though, LCC put across a very professional front, and indeed have some very competent people working there, but that gives the impression of a much more deep-pocketed organisation than they really are. It's a difficult spot for them in some ways - £40 is a fair bit for most people to spend on a membership that brings rather little in direct benefit, but equally with only 10,000 members it's not a massive budget to work with.
The local groups are for the most part small, often between 5 and 10 active volunteers, so when any one junction gets put under the microscope, it falls very much to individuals in those groups to make positive changes happen. Very often on a given street or junction improvement, there will be just one or two people driving the agenda to make it better. They'll rally others round, but still.. it's rather a case of, if you don't like what they're doing, get involved & make something better happen instead.
And while yes, there are major hazards every few miles, you don't need to fix every last one to make things much easier for novices & knock a lot of those <3-mile car trips on the head. Novices by definition are unlikely to be cycling from one end of town to the other. The "Hackney solution" of getting through traffic on the back streets works well for that in a lot of cases, but on the CS2 route there just aren't any minor-road alternatives - that in itself is a strong argument for segregation.
In the case of, say, CS8, the argument is weaker - if you want a family ride along that way, the river path works just as well (if and when they finish the Battersea Power Station redevelopment, at least).