No. There is no e6 variant in 5e.
I've met people who worked out how to level beyond 20th, calculating XP tables, proficiency bonuses, etc., but I haven't seen anything for capping the game below 20th. Instead, I have experienced a few surprising changes to NPC structure that may already solve your problem.
First of all, for your demands, there is a carrot stick approach.
Let's Begin with the Carrot
The DMG presents sections on Pages 228 to 231 covering options including Marks of Prestige, Parcels of Land, Titles, and Training, which include things like granting proficiency in a skill or a new feat.
These are the tools a king or empire can use to influence high level characters to do things like stopping a peasant revolt, under the conditions that a minimum number of lives are lost. You could even make a short chart indicating cumulative merits, like, "no lives lost, total success" vs "few lost lives", "major political/military strategic person/place destroyed" vs. "major political rival/fortress captured". Then dole out better titles and awards (cash, treasure, medals, etc.) in exchange.
You could even earmark certain grand awards on a tier, like its own level system or "merit badge ranking" system, where a character can become an earl if they earn medals A,B, and X and complete 2 minor, and 1 major missions/boons, etc.
All the tools to do this are right there in the DMG pages I mentioned, and all you need to do is remember that you control the politicians, their titles, ranks, and coffers.
The average player character, given a choice between low XP awards for blasting an irritating peasant, and wedding a lady of the court, marrying into a famous house, becoming knighted, gaining a keep or castle, or getting those precious weeks/months of training to learn that precious feat without having to level...
They are almost always going to take the boons.
Now for the Stick
When I first got into 5e, I began to notice "these orcs were not the same as grandpa's orcs". The NPCs aren't zero level. Even in a small number, like 5 or 6, a few CR 1-4 NPCs can lay waste to a player character.
They can get you when you are sleeping in the Inn, or drunk and wandering down the ally. They can get you when you are shopping or crossing that draw bridge, to go prove how tough you are, only to discover what goes wrong when the NPCs all have just enough HP to survive the first blast of fireball. How high level does an NPC have to be to survive 8d6?
When you have a group of 6-30 NPCs, not as high as you think. About half will make their saving throws. 8d6 averages about 24-26. But saving for half only requires NPCs to have 13-14 hit points to survive. A bunch of CR 2-3 peasants, captains, and soldiers would never be seen in AD&D, but in 5e, they are all over the place. Even the local priest, CR 2, has 27 hit points - that's like two fireballs in a row and he's still barking orders at his zealots!
Have a number of knights guarding the gates at a rival Duke's palace in the middle of a War of Roses? All 8+ of those guys in armor have 52 hit points a piece, and are only CR 3. Sure, the PC's think they are Guards (11HP, CR 1/8) but the PCs don't need to know that.
If you want to get really mean, there are tv series like Spartacus, that illustrate whole pits full of Gladiators. Literally, a small army of gladiators, ranging from 15-300 men. How bad could that be?
The lowly CR 5 Gladiators only have 112 hit points a piece. Wait. What? But commoners are 4 hit points, and it says "peasants, serfs, slaves...hermits". Sure. But Gladiators are also slaves. And hermits are also Priests and Spell Casters.
By sticking with CR 1-5 NPCs, and using them in small bundles, you can keep the peasants far lower level that say, a 14th level Warlock or 12th level Barbarian, but even 5-6 of them could drop your most obstinate PC.
I had a very high level 5e character, something like +14 to hit, multiple attacks, spells, sneak attacks - even had fireballs. And he didn't stand a chance against 8 human "guards" at the gates.
You, as the DM don't just command the gates. You command the whole revolution. The whole kingdom, and all its neighbors. You have literally millions of people to draw from and among them, can easily cluster dozens of mediocre CR 2-5 traditional NPCs (like knights, gladiators, and cult fanatics) to surround and imprison/execute uppity player characters.
Balancing Stick and Carrot
When you demonstrate the stick, it adds weight to the otherwise "fluff" aspect of the carrot options that don't seem as useful. When the players see the power wielded, then they will start to respect the value of being able to walk about those "walls of meat" guards because of their Medals of Valor, or being awarded with NPC servants that can actually do things by means of land and title. A Paladin having command of a squadron of Knights, or a Fighter being gifted their own Ludus (Gladiator training hall) with one or more gladiators to command has more impact than the CR 3-5 implies.
Letting your players build into the world, like actual buildings, territories, alliances, and relationships bolsters the value and personal interest in ways that leveling up doesn't. Meteor Swarm is nice, but many will find a Palace is better.