Some things to consider:
1.) Monks can do damage earlier than other classes with their high mobility. I often get into melee a round before the other members of my party, and usually go before the heavy weapon hitters.
2.) Part of the damage others deal is monk damage. When we stun, and someone gets advantage, and hits because of that advantage, that is our damage.
3.) When we Flurry, we make 4 attacks. We can do this at 5th level. Each of our attacks may deal less damage than the heavy hitters, but we lose less to overkill as well.
So, DPR may be lower, but effective damage per combat is going to be closer, if not better, than slow moving heavy hitters.
My 8th level monk has a flame tongue short sword. He deals (d6+5 piercing + 2d6 fire) on his multi-attacks and d6+6 (blunt force trauma) on his unarmed strikes (Insignia of Claws). Under Tasha's rules, I could do better - but I find that is plenty effective. He also is a Vhuman and has Hex from Magic Adept (which he primarily uses out of combat - to give someone disadvantage on their chance to detect him when he sneaks or pick pockets). However, when he snuck up on a dragon, he managed to hex it and get off a dozen strikes on it for 12 d6 additional damage, and he also uses to pump up damage when he is forced to use a bow to attack a flying enemy.
Regardless, combats generally go like this:
We encounter enemies and start off at a distance of over 30 feet. I usually go early, charge into the fray and get at least 2 chances to stun something nasty. I often have enough movement to retreat, or I use a bit of Ki to take the Dodge action and absorb a bunch of attacks and keep the enemy contained.
Sometimes I get the drop on the enemies when they're poorly positioned and (being an open hand monk) I am able to send them off a cliff, into lava, etc... The DM had intended for one BBEG to be a recurring villain - but I snuck up invisible on it when it was giving an inspirational speech from the balcony of his lava surrounded fortress.... Hit, stun, flurry hit push 15 over the railing and down into lava near a wall and 90 feet from any shore. I highly respect the DM for not cheating to save the BBEG and to let the story unfold as it did. The inspirational speech devolved into panic from his troops... Fond memory.