I don't think a new series would need the original trilogy to have new characters to be good. But I want to see a post-Empire Star Wars Universe and what it would be like. Like do the Crime Syndicates try to fill the Empire's void at all and become a major threat? Or as Roose mentioned, do the Chiss Ascendancy reveal themselves as a major player and reveal themselves to the rest of the galaxy.
I think the OT characters would be interesting supporting characters (Luke especially as a periodically appearing wandering Jedi Knight Errant) and background movers and shakers; like I wouldn't mind having some new characters work as part of Han's shipping business, since I presume he's probably just mixed smuggler skills with legal work in a free but still very dangerous Galaxy, and Leia could easily be the Senate leader of the intelligence community that oversees other heroes.
And we've got numerous little hints at fun possible adversaries: at least some Imperial remnants are still active after Jakku, like Moff Rand, the Galaxy has two successor states to the CIS, whole sections of the Galaxy seem to not be part of the New Republic, and there's a
pirate nation state with a Super Star Destroyer. It's not just the GFFA without the Empire, it's the Wild Wild West in Space.
By the by, I've had some pseudo head-canon going on for what an post-OT series would be: a team of peacekeepers approved by all the new nation states to track down ex-Imperials, notable criminals, and charged to secure the space lanes. It's a chaotic mess with cycles of different types of big bad guys: the Pirates of the Latitudes need to be either contained or crippled, Imperial diehards try to hold onto a few obscure military bases (and the few that escape are hinted to joint the FO), new antagonists play a more subtle game (imagine Warlord Zsinj from Legends reimagined for the new canon), and just a general rogues gallery of threats threatens the peace of the Galaxy.
Oh, and since I'm a Rey Skywalker supporter, the main star would be ex-Imperial Agent Mara Jade, as she struggle so it has emitting her past allegiance was horrible and considers taking that Skywlaker fellow up on his force training proposal...