Quick summary of the thread
* We don't need segregated infrastructure
* Oh yes we do (sometimes)
I think I'm in the sometimes camp.
A couple of sometimes ...
For busy A roads between towns where there is no parallel series of minor roads, a cycle path alongside can be a boon. It's OK for me as I don't mind the odd stretch of busy road. But where there's a decent cycleway I'll use it - it reduces stress levels. For less confident riders it opens up a route. I rode one recently - the A413 Winslow - Buckingham. It was generally OK-ish
Shared use paths - yes you are reduced to walking pace and yes you may have to deal with dogs on extending leads, people with headphones and so on. Nothing wrong with that provided you adjust your expectations accordingly, slow down and treat people with respect. The shared use signs are a reminder to everyone that yes, cyclists really are allowed here.
That's just speaking personally as a relatively confident rider.
I think there's an issue that the range of requirements for a cycle path is wider than that for a road. Bear with me while I expand on that.
Drivers are broadly expected all to drive in the same way - approaching the speed limit (but not beyond it) on clear roads, more cautiously when conditions demand it. OK, larger vehicles, those towing caravans, and so on, shouldn't be doing the same limits, but the variation in speeds is, what, less than two times in most conditions (and lots of drivers still find it stressful dealing with those going faster or slower). A certain level of confidence is demanded to pass the test and get on the roads in the first place.
Cyclists, on the other hand, range from kids and some adults barely managing 5mph up to club riders sustaining perhaps 25mph, which is five times. And those trying to cover distance are looking for clear, open routes, whereas potterers and those nipping to the shops find it much easier to handle a more stop-start style, mix it with pedestrians, etc. It's really not feasible to do a 100-mile ride on a shared-use path system, at least when there are people there to share the usage.
For drivers, authorities deal with that by building a variety of road types - roads into town with lots of traffic lights, bypasses, motorways, etc. For cyclists, it's one-size-fits-all, in spite of what I have argued is a wider variation of needs. Yes, I know 70mph on a motorway is seven times 10mph in a car around town, but the driver doing 10mph around town usually could do 70mph on the right road, and that's what's different for cycling. The person doing 5 or 6mph probably couldn't go any faster when the only user on a converted railway line.
So it doesn't surprise me that designing one-size-fits all cycle facilities gets a range of reactions, including the perfectly reasonable choice (in many situations) to prefer the road.
I'm in the sometimes camp too. No objection to cycle paths, but I do exactly what I do when I'm driving, and choose a combination of safety, convenience, likely travel time and how I'm feeling today when selecting a route. Just as, only today, I turned off a statistically-safer motorway onto therefore-riskier A road earlier than I needed to, because it suited me.