Like others have said, the cycle route through the park has very clear sight lines along most of the way, so you can go faster than 8mph perfectly safely, if you do watch out for dogs, people on the paths at right angles, and gormless cyclists who are too busy reading an SMS on their phones or similar (I get this once every few months).
An 8mph limit verges on the ridiculous, and I don't think any guidelines anywhere suggest that it's a sensible limit for cycles on any sort of route. If you need a speed that low, it's probably not appropriate to have a route which is so close to pedestrians at all, but that's not really the problem here.
It's an important route because it feeds directly onto Cedars Road, which itself is pretty directly in line with Chelsea Bridge, and in the other direction is fed from various southerly directions, and various useful and important routes, including CS7.
Click for a bigger and more extensive map This map shows the Clapham Common path we're talking about in
Red, although the bit along the Windmill Drive (the middle of the three red lines) is actually a normal road, and covered by the normal highway laws. It's also frequently used by car drivers doing a lot more than 8mph, and fairly often more than 30mph.
The
Magenta lines show some feeds into this path from the south (including CS7), and the
Cyan show the direction that many cyclists follow north of it.
In
Black I've tried to show some alternative routes. The one to the West has some fairly nasty junctions and the traffic levels on that road are very high. It's not a nice road to cycle on, and you'd avoid it, since it's significantly more dangerous than going across the Common and along Queenstown Road.
The routes to the East of the common isn't as bad as the Western one, but still does have some iffy junctions, and unless they've resurfaced a lot of it recently has some very bad surfaces, to the extent that if I couldn't cross the Common, I'd seriously consider using the Western route rather than this one.
All of the roads immediately around the Common are very busy at most times, and not very pleasant to use at all.
The route across the Common is useful, convenient and important to London cyclists. As far as I'm aware there isn't any significant history of problems, certainly not compared to many of the roads in London, which are generally far less friendly for both pedestrians and cyclists. It seems very odd the effort that seems to be going into discouraging cyclists from using what admittedly isn't a perfect facility, but which is a damned site better than many. The local councils should be improving it and encouraging cyclists to use it. Pushing cyclists off of the common and onto the surrounding roads is certain to increase the number of collisions between cyclists and motor vehicles, with consequent effects on injuries and fatalities.