A grappling monk would be great for roll-playing. Descriptive martial art combat! Clinch and strike. Grab and throw (grapple then knock down).
Unfortunately, it is weak grappler compared to other class choices. I think the best chance is to get a level of rogue so you can "expertise" athletics.
There is an open-hand technique to knock prone as part of furry of blows.
Otherwise you might need some house-rules to help bridge the gap to make it on-par with Barbarian.
On the monk side, you could add (or replace) some class features - limited by Ki points for balance:
maybe a "grappling strike" (ki point to apply grapple condition after a successful strike - or at least a grapple check)
(note: this help action economy so you can strike/grapple then do furry of blows and knock prone in one turn - which would role-play as a brutal take-down or throw.)
The grapple feat is considered under powered. It could be boosted without unbalancing.
* The 'advantage' feature is of minimal use when the most common strategy is to knock the opponent prone. Since the opponent can replace any of it's attacks to attempt a shove (to escape the grapple), it is preferable to get the opponent prone. So maybe add a feature to "counter" a shove: if a shove misses, the attacker is knocked prone (very jujitsu/aikido flavor)
* Let the feat give the monk some love: Allow wisdom bonus to grapple checks. (which makes total sense if you have ever grappled with a seemingly weak & slow elderly jujitsu master -- you are on the ground before you can think!)
Those would make a difference balancing the monk grappler with another grappler build and give meaning to the grappler feat (without making a Barbarian/Bard/Rogue grappler more powerful)