Let’s take this apart and see.
Grapple : using the Attack Action you may make a grapple check. If you have Extra Attack you may use this instead of one attack.
Martial Arts: if you make an attack with a monk weapon or an Unarmed Strike you may make an additional Unarmed Strike as a bonus action. (I don’t think he can do what you have described RAW)
But....
Flurry of Blows: after using the attack action (which could be a grapple) you may spend one Ki point and make two Unarmed Strikes as a bonus action.
And....
When a Path of the Open Hand monk hits an opponent with an Unarmed Strike as part of a Flurry of Blows the monk can make the target roll a Dexterity save or be knocked prone. Alternatively the monk can cause the target make a Strength save or be shoved up to 15 feet away. A third alternative the monk can choose to deny the target the use of it’s Reactions for one turn. (No save IIRC)
The way I read it the monk could grapple the target (Attack action and possibly reposition) then spend a Ki point to Flurry. Hit the grappled target ( or another in reach) potentially knocking them down. (Dex save) Then Attack the downed target with advantage. At this time the target makes a Strength save or be shoved 15 feet while prone.
The picture I get from the OPs description is that he isn't spending Ki. He's avoiding the expected cost. He should be able to do something like this (from 5th level) -
Attack
Extra attack > Grapple
Spend ki for a Flurry with a bonus action
Trigger Open Hand for a saving throw to prone
The grapple is superfluous there as he can also, by RAW -
Attack
Extra Attack
Spend Ki to Flurry
Trigger Open Hand for a saving throw to prone, possibly twice depending on how you read "Whenever..."
What he can't do by RAW is -
Attack
Extra attack > grapple
Bonus action unarmed attack somehow triggers Open Hand
That's specifically excluded. Maybe in part because,
given the grapple makes the target's speed 0, it can't get up, and maybe in part because it also then becomes inexplicable why he can't -
Attack with unarmed
Trigger Open Hand (how?)
Extra attack with unarmed
Trigger Open Hand (!)
Patient Defense, Step of the Wind or Flurry vs now prone.
The point is, a Monk can already do something pretty powerful with Open Hand (Attack, Extra attack > Grapple, 1 Ki to Flurry, trigger Open Hand up to twice to prone: target is now down with speed = 0)! It feels like they could just play RAW and have a big slice of the fun they want to have.