If you get ahead in culture you can really rack up the free envoys. And you should basically always be using the 2-for-1 first envoy card if you're going to put the first envoy there. I mean, even if your next policy card was 10 turns away, you're giving up maybe 20 points of envoys for a free one. Or if you're running the dollars per envoy card, maybe it's 100 gold that you give up, but if it means getting to the 3rd envoy on the city-state faster, it can definitely pay off. And if you put even a modest amount of effort to completing quests, that can really add up.
One thing to keep in mind too is that you get new quests each era. So as you're getting close to the end of an era, that's a great time to run through the CS list and see what makes sense to get done. Or if you can't complete a mission, it can also pay off to purposefully fail it. So if I have 2 or 3 city-states asking me to get the get the eureka for a tech that I know I'll never get, it might make sense to research it anyways so that next era they give me a new challenge. Or a lot of times, if I have a couple city-states asking me to build a spearman, it might be worth it to basically spend the couple hundred gold for 2 envoys.