First, the level 11 feature of monks doesn't have to carry a full extra attack's worth of damage.
Monks damage comes from a gradual increase in MA die and more Ki to spend on Flurry over a day, just like how a Rogue's damage comes from their sneak attack getting bigger.
OTOH, it is true that a bump at 5/11/17 is reasonable. And there is evidence that the Monk's damage output falls off at and past 11, as the monk reaches saturation of flurry of blows damage.
My solution was 3 fold.
Martial Arts: (change)
You can use Dexterity instead of Strength for the Attack Rolls of your your unarmed strikes and monk weapons when not wearing armor. You can add both your Dexterity and Strength to the damage rolls of your unarmed strikes and monk weapons.
One with Ki:
Starting at level 11, the first Ki you expend in a round is free, unless that Ki uses a monk class or subclass feature to heals or restores HP. In addition, once on your turn when you use a bonus action to spend Ki, you may take an additional different bonus action on your turn; it may not be the same bonus action twice.
Starting at level 17, when you spend Ki as a bonus action, you may do the the bonus action twice. Each use still costs Ki, and you can still do an additional different bonus action.
Perfect Self: (change)
The first 2 Ki you spend in a round is free, even if you use the Ki on a monk class of subclass ability to restore or heal HP. If you miss with an attack or an attack hits you, you can expend 1 Ki to cause that attack to be rerolled; if you don't like the result, you can spend more Ki to make it reroll further.
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The Martial Arts change makes Dex+Str be a damage dealing monk build. You have to be unarmored, so low Wisdom+Dex still sucks. But a 12 strength (or higher) gives you a damage buff. Equally important to me, it encourages Monks to be buff, without making it dominant.
One with Ki lets the monk flurry all day long, as they tend to do. It also lets them do other things with Ki without tanking their damage per round. A monk that flurries on every attack at level 11 isn't falling behind the damage curve.
At level 17, the ability to double-flurry comes online. 6 attacks for 1d10+dex+strength is nothing to sneeze at.
You also get to do gonzo things like double dash with +30 ft movement speed, moving 180 feet, then doing a 6 attack flurry (price: 3 Ki, including the free one).
The lack of pressure on your Ki from your large pool and your level 11 ability returns.
At level 20, we change Perfect Self from "you always have 4 Ki" to "the first 2 points you spend is free", and add in a new way to spend Ki like crazy. The idea of "you just don't miss" and "you just aren't hit" until you run out of Ki is fun.
I also removed the "no using this for healing". If your level 20 monk wants to do non-combat healing of everyone after a battle, have fun.
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The "burst" DPS of a flurry-using monk at 11 is fine; I just made it baseline. Now you are spending Ki on dodging, dashing, stunning, or your subclass abilities rather than hitting baseline DPS.
Your ability to invest in Strength to boost damage (instead of Wis or a feat) also provides a way to keep up with the Joneses.
By level 17 your DPS is starting to fall behind; so we go a bit gonzo and allow a double-flurry (and a double-dash, and a double-dodge which is useless but hey)