Family Feud is a buggy quest but its pretty great since if gives u 10 relation with the quest giver when completed while only losing 5 with the hurt party. Sadly the quest is one of the more buggy. Thankfully, I have 2 solutions that work on two of the issues I've encountered.

If when fighting the npc when you finish him and his thugs he talks to you like nothing happens... The only solution I could find was to reload to a previous save and savescum the conversation way to finish the quest. When talking to him click the option that mentions the law and save scum until you get the outcome that results in no fight. Or of course you can turn the npc you're escorting over.

The other bug I encountered had it so the npc I was escorting would not enter the town with me and therefore when I talked to the npc I was at the village for he would talk to me normally and the quest dialogue would not trigger. To fix this go to the town screen but dont actually enter the town instance, click on the clan button, kick the npc you were escorting from your clan. Now he will be residing in the town. Next, talk to that npc in the town, and ask him to lead you to the npc you were at the town for, follow him to the npc, and then talk to him when they are close together. When you do this the quest can then be completed.

Note: if when you kick the npc out of your clan he is not in the town; exit then reenter the town menu screen and he should be in there now.

Edit: Another way to leave the NPC in the town is to click at the arrow thing in the top left and select the NPC to leave them in town. Quicker than going to the town menu. Then click on that npc in the list of town npc's and ask them to guide you to the target npc. Thanks - LagMasterShaun

I hope this helps, if there are any other bugs with the quest that I did not mention here leave them as a comment and hopefully we can crowdsource a fix. I will add and credit any other bugs / fixes.