On Patch 9.1, it turns out that there's really no "low effort" way to quickly reach maximum renown. There are, however, some quick (albeit very grind-intensive) ways to do so, though.
After Patch 9.2, there is a much easier way, though, as Nzall explains.
Skip to Renown 60
If you have at least one character on your account that has 80 renown with any covenant, you can buy the Broker Mark of Distinction, available for purchase from Au'Dara, which is Blizzard account-bound and instantly jumps your Renown to 60 on your current covenant choice.
Torghast, Layers 9/12
The quickest method is to spam layer 9 Torghast runs. The layer's end boss has a high chance of dropping an Emblem of Renown, and thus by very quickly travelling to each floor's end boss, killing it, then quickly continuing onward to the next floor will net you a lot of Renown per hour.
This requires decent gear (around 200 item level) to be efficient and to have a lower chance of dying, though. If you don't have decent gear, you should look elsewhere for Renown, as you'll either continuously die to the somewhat difficult enemies when running solo, or do little damage such that the runs take too long to complete.
Weekly World Bosses/Quests
You get an Emblem of Renown for defeating the current world bosses for the week. One is located in one of the Shadowlands realms, while another is in the Maw.
Within the Maw, there are covenant assaults you can participate in which take a moderate amount of time to complete. There are two covenant assaults per week, and each will grant you an Emblem of Renown.
Additionally, there are two weekly quests you can get at your covenant sanctum, one that requires you collect 1,000 anima, and one that requires you collect 20 souls from the Maw. The soul collection quest is also repeatable.
Altogether, these account for 6 renown if you complete them all before the weekly reset on Tuesday.
Looking For Raid (LFR) Runs
Running the LFR queue for the current raid tier seems to be fairly quick, and if the gear at that level is beneficial, you do have a chance of getting helpful upgrades. Each boss you kill has a high chance (if not 100% chance) to drop an Emblem of Renown. Unfortunately, this is heavily dependent upon how decent the group you've queued into is, as inexperienced players might die and delay how quickly you complete the run.
This is also gated as you can only get loot from a specific raid boss on a specific difficulty once per week, so for the current raid tier (Sanctum of Domination) you can only get 10 Renown when running LFR. This is largely recommended for players who would have a chance of getting a gear upgrade from the runs, and not as a primary way of gaining renown.
Covenant Campaign / Story Quests
Blizzard's most recent changes to Renown catchup have allowed you to put less emphasis on your covenant campaign, but the campaign itself is still a decent way to get Renown. Every couple of quest completions you'll get one Emblem of Renown. The downside to this method is that it's fairly slow for some covenants, and if you're like me and already familiar with the story, this is extraordinarily boring. At least it's easy and somewhat consistent.