Sophie and I ran together in a self-supported style carrying our own water and food/gels. The weather gods were sadly against us, but it did make for empty roads and sites. We started at 750am from Lambeth North tube station, which on a sunday morning is about the earliest you can get there. Starting early allows you to enjoy some of the prime spots earlier enough in the day before the shops open and the tourists arrive.
The route I had pre-planned and intended to take in a few parks along the way and not always to take the very shortest route along the main thoroughfares (eg from Marylebone station to Euston Road we stuck to the back streets south of the main road). We also added in a couple of minor detours to take in some additional non-monopoly board sites, such as Buckingham Palace (you could otherwise save around 1km between Pall Mall and Piccadilly). You really do get to see all parts of London on a route like this, along with plenty of unexpected surprises.
Whilst some "places" are very specific, others are simply roads. I took the view that you had to visit each of these places, but not necessarily traverse the full length of the route.