So, some spoilers for the Chronicles of the Gatekeeper below, if that's not a thing you want in your life.

I know that even a written adventure can't spell everything out for you, and that a lot of a topic like this is going to be covered by how the GM chooses to flavor it or the approach PCs take. But given the impact it can have on the future of the city, I feel like I - as a relatively novice GM - need some more to go on, both to run the thing, and to help my players realize it's a salient part of life in the city that should be resolved. Any advice?

Go to the source. Hard to top a scenario sketched by the Bard himself. This game accommodates fine literature as well as pulp

Yeah, street scuffles, honor duels, the occasional aligned shopkeeper badmouthing the other side. If you make it a thing going on, the players will naturally be drawn into at least the atmosphere, and recognize the major parties when it happens.

I'm running the same adventure and we've just finished our first session in Quolas. Just wanted to share with you how it played out with my players:

They arrived in the city and began brainstorming a plan to get to the reeve. While my Seeker and Consular are engrossed in strategy, my Sentinel notices a man being beaten by the city guard. He marches over, followed shortly by the other two, and tries to calm the situation. After a painful number of failures and threats the Sentinel finally flashes his weapon from underneath his cloak, prompting the two guards to take off. The man they saved is so grateful that he invites them back to his bakery for some free food. They ask him about the reeve, the city, Suljo Warde, etc. During this conversation he discusses the feud between the two families, and says that if anyone in the city would have been important enough to talk to Warde about his plans when he left the city, it would have been them. He tells them what level of the city the estates of the two families can be found. On their way up they came across the Pillar of Ancients about to be destroyed. The encounter came to a head when my Seeker took up a hidden position, pulled out her Corellian Bow and aimed an explosive shot at a nearby light source, to try and distract the guards. She failed her roll with a despair, so her arrow flew high and hit a nearby pod that is a Sathari lounge. It caused the whole crowd, including the guards, to rush over to see what had happened. During the commotion the players heard the name Pathran Helshar mentioned a few times as a possible cause for the explosion. After that they headed to the Helshar estate and met with Koh. After some negotiations and questioning, she told them what she knew about Warde and revealed that Pathran is her son. The players were adamant about meeting him and the Consular succeeded on her Negotiate check to try and strike a deal if Koh would arrange a meeting with her son. She told them if they could get Odir Tumris to the negotiating table to try and broker peace, she would arrange the meeting. They immediately went to the Tumris estate, and after navigating some hostile lackeys of Tumris in a bar fight (on of them smacked my Seeker on the rear end, which she did not care for) Odir invited them back to his office. This scene played out in about a quarter of the time because the Consular got a Triumph on her Charm, Leadership, and Negotiate rolls, Odir Tumris may be in love with her. He has agreed to meet about brokering a truce and possibly a means to overthrow the reeve.

That's what happened in our session, but the important thing is that none of it was railroaded. I presented information and situations to the players, and I reacted to their actions and even mistakes. The reason the adventure doesn't give you a clear path is because the players are meant to forge their own trail and just pick up on the information and clues you give them. I knew that the feuding families needed to be resolved, and I also knew I needed to introduce the resistance, so I waited for my players to do the rights things and ask the right questions so that I could insert that information and then let them run with it. If you give your players the information about the feuding families, and they don't think that the situation is a priority then let them make that mistake and describe the destruction that begins almost immediately after they leave the city. Some of my players' favorite storytelling moments have been when they made mistakes, or disregarded a piece of information, because it makes the world feel alive and like it goes on without them. Don't be afraid to let your players fail at a storyline, as long as its a side story.

Thanks for the advice! Now I just have to convince them it's safe to wander the streets after attacking the reeve's guards.

Give a lot of time, even a full session or two to get aquinted with the city, its citizens and the alien society. Dont lead them into major plot points yet, but make sure they get to know NPCs and agendas.