That map makes sense if there is no traffic backed up on the South Circular from Forest Hill to well back beyond the Horniman. At peaks times it often is. It does move quite well, but some people are congenitally incapable of sitting in traffic and will always take an alternative route that FEELS quicker because they are moving.
I have a feeling that people who regularly drive south along the South Circular have it firmly fixed in their heads that they must avoid the junction at Forest Hill at all costs. So if you're coming from Peckham direction, up Forest Hill Road, taking either Woodvale or Honor Oak Road all the way down to the South Circular doesn't seem to make sense. Psychologically, creeping and crawling down Devonshire Road seems like the correct choice, because you have avoided Forest Hill.
Personally, I'd always sit in a bit of slow moving traffic through FH rather than take that rat run. I don't have the nerve for it, and I like my wing mirrors attached to my car. And I suspect it's not much quicker. But in times past, when the sequencing of the lights at FH was much worse (I'm thinking about ten years ago) and there was less traffic on Devonshire in general, it did make sense. Once a rat run is established in driver folk lore, it's hard to persuade people it's actually slower than taking the main road.