I bumped into a couple last year near St Martha’s. One on a road bike the other on a hybrid step through thing. They were walking down a section of bridleway which is a blast on an MTB. Fast, loose, rocky, sandy with a few steps to launch off. They were fairly unimpressed having kicked up the route online as being suitable for road bikes.
I haven’t ridden the lenght of the Downs Link but I’m not convinced that top section would be rideable on a road bike with raid wheels/ tyres.
This was the the bit from St Martha’s Hill down to Lockner Farm. Once you cross over the road, it’s a bit if a mud fest and then into a sandy heath. Not sure about the rest as I usually head the opposite way from there. That top bit though, seems a bit much for a road bike?
You could skip that top section easily enough. I’m sure someone with more experience will be along shortly to advise on the rest.