I currently have 2 level 120s and I am pretty good at making sure I don't miss any emissaries. My favourite emissary reward is the 2000 gold reward.

The thing about emissary quests is that they're a good baseline around which to structure your gameplay - it's the way they have been designed. The active emissaries end up influencing which zone I'll go visit - I'll pick a group of quests based on a combination of the quest rewards, whether the quests are fun or tedious, synergies - where a quest doubles up for multiple objectives (eg Tortollan/Champion of Azeroth/invasions) and the amount of travelling I have to do in the zone. While I am in the zone I'll often do some opportunistic activities - extra WQs if they're convenient, fishing, material gathering, archaeology, pet battles, achievements etc.

For a chilled/relaxing game play experience, I find this is perfect. Doing all your emissaries is minimal effort and ensures that you keep your Azerite ticking over while bringing in a steady stream of war resources to drive the mission table and loot coins. The rep gained for paragon chests is also great because those reward a substantial gold and war resource bonus.

As I said, right now I have only 2 level 120s and doing the emissaries on both is a breeze. I think once I get to 4 x 120s I'll stop worrying about doing all the emissaries and start focussing on the best value ones - like the 2K gold reward, or, going for them when they line up with Tortollan/Champions or an invasion eg doing the Zuldazar invasion when Honorbound + Zandalari emissaries are up means I complete 3 objectives for the effort of one.