Let’s do this! Can you do it without variant human? If so let’s start comparing with rogue.
For a quick comparison, let's look at level 5, since that should be a comparative spike for monk, since they get extra attack, whereas rogue just gets another d6. We'll assume 8 rounds per short rest, which has been the standard in this discussion.
Let's assume the monk is using all ki on flurry of blows. Since the monk is using all their ki to attack, they are not going to be able to use Patient Defense or Step of the Wind, so let's assume their subclass features are used to help them survive (e.g., via Drunken Technique, or Open Hand Technique taking away the enemy's reaction). Of course neither of these are available on non-Flurry turns. We could try to model occasional advantage from an open hand monk also knocking an enemy prone, but for simplicity I'll instead try to lowball the proportion of the time the rogue has advantage. The monk boosts DEX at 4th.
On the other side, let's take an elf arcane trickster using a rapier with booming blade, getting advantage from a familiar half the time (that's a bit low, in my experience, but we want to lowball this a bit to compensate for the monk also occasionally having advantage). They take elven accuracy at 4th which also boosts DEX. The rogue's bonus action is free, so they can disengage every round.
Monk:
2 staff attacks at 65% to hit: 2 * (4.5 + 4) * 0.65 + 2 * 4.5 * 0.05 = 11.5 expected damage
Unarmed strike at 65% to hit: (3.5 + 4) * 0.65 + 3.5 * 0.05 = 5.05 expected damage
Second unarmed strike: (5/8) * 5.05 = 3.16 damage averaged over rounds
Total: 19.7
Rogue:
1 rapier attack,
* 65% to hit half the time, 96% to hit half the time (with elven accuracy), for 80% on average
* 5% chance to crit half the time, 14% chance to crit half the time, for 10% on average
* Expected damage: (2*4.5 + 3*3.5 + 4) * 0.80 + (2*4.5 + 3*3.5) * 0.10 = 20.8
So they're roughly on par at 5th (a relatively favorable level for the monk), since a swing of 1.1 DPR is easily within approximation error.
However, when not attacking in melee, the rogue has the clear edge: they are a 2nd level spellcaster, with 3 spell slots per day (we could likely do more damage with a swashbuckler, two more skill proficiencies, and expertise in two skills, and also have the ability to do nearly the same damage at range. If the monk has to attack at range, they are making two shortbow attacks for d6+4.
On survivability, the rogue has a damage reduction ability (uncanny dodge) which likely prevents more damage than Slow Fall, Deflect Missiles, and the monk's +1 AC combined, and can disengage every round, whereas the monk can only do so 5/8 of the time.
Furthermore, the rogue is some of the time doing some "soft control" via booming blade (or is doing another 2d8 damage).
As they go up in level, both the rogue and the monk fall off compared to the ranger, fighter, and paladin, but in tier 2, the arcane trickster in particular gets some very nice toys in the form of more spells, spell slots (some of which can be used to boost damage if desired), and a nice spell-boosting feature (magical ambush), as well as the bonus feat that all rogues get at 10th. The monk gets more ki, and hit a point when they can flurry every round if they want to, but that has less offensive impact than sneak attack dice. They also get more movement, a poorly designed anti-charm/frightened feature, and a subclass feature, most of which are either a survivability boost (helping them be less behind in that area), a mobility boost, or yet another way to spend ki. Shadow at least gets something neat which doesn't use ki. The monk does get magic fists, which is definitely useful in a campaign where PCs are not getting suitable magic weapons; but that's not the norm IME.