Why do you think Stannis focusses on the direct relationship between king and individual subject, rather than king-vassals? And why does he focus on centralising the Crown as the seat of justice, rather than allowing the lords to distribute their own justice? Is having Mel and Davos (both “lowborn”) around part of it?

I don’t know. It seems to be strongly rooted, so it’s not a recent idea – and given Stannis’ reference to his various readings,my guess is that this is an idea he developed when he was being educated about Aegon’s Conquest, the Dance of the Dragons, Daeron’s Conquest, etc. After all, it’s hard not to read about Jaehaerys I, Viserys II, Aegon V, etc. and not see that you need a centralizing monarch to enforce justice.