My proposal is to embrace the MAD. Monks are perfecting their selves. Run with it.
Martial Arts (modification)
* You can use Dexterity instead of Strength for Attack rolls of your unarmed strikes and monk weapons. When you strike with an unarmed strike or monk weapon, you can add both your dexterity and one half (rounded up) of your strength bonus to the damage roll instead of your strength or dexterity bonus.
Perfection of Mind and Body (additional class feature)
At level 4, 8, 12, 16 and 19 the Monk's efforts to perfect their mind and body pay off. They gain a +1 bonus to their strength, dexterity, constitution and wisdom ability scores. If this would cause their ability score to exceed 20, they can instead assign the increase to any ability score of their choice.
Increases from this feature and from your ability score improvement at the same level (or feats) cannot cause an ability to increase by more than +2; any excess should be reassigned to an alternative ability score.
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This helps monks scale. A level 20 monk now has +5 dex/str/con/wis over the existing monk, enough to cap many of those stats at 20, plus their usual ASI points.
If they have 20 str/dex, their monk damage is 1d10+8, which is a nice increase. They are encouraged to be not-weak at low levels (12 strength is worth +1 to damage on all of their attacks). If they start with 13 strength, they'll end up with 18 at level 19 for no further investment.
I might also be tempted to nerf stuning strike and give out more toys.