I would prefer to add feats to the already existing structure. (In my mind, one of 5e's flaws is publishing a bunch of feats but then providing very few choice-points at which a feat can be chosen.)
I think the background features could provide a lot of ways to engage with the in-game world and have the PCs be built as part of the world (rather than being divorced from it).
Also, part of how I see that concept being expanded (instead of being cut) would be to flesh things out more so as to avoid conflicts like the one you cited. In other games I play, it would be understood that a character's reputation with a faction (i.e. the Folk Hero's "common folk") would be dependent upon proximity, fame, or other such factors (that could be explored through play).
*Edit to Provide an Example: Maybe in the Folk Hero's hometown, the feature works automatically. However, in a different kingdom, it may be less-likely that the hero's story is known or recognized, so the DM asks for some sort of roll. Later, if the hero gains more fame, perhaps the area for which no roll is required grows larger. In contrast, if the "Folk Hero" builds a reputation as being a selfish murder-hobo that robs shopkeepers, perhaps some of that benefit starts to go away -even in the hero's home area. I see it as a narrative negotiation between the player and the DM, as well as a resource that may help the group approach an adventure in a different way than strictly planned by the book.>
It's an aspect of 5E that has/had a lot of room to grow, but it was largely left to wither and die after the initial core books.
I had hoped that some of the promised "modular design" might further explore those aspects of play.