Pick a quarterstaff with Topple for instance, getting advantage on flurry of blows will probably do more damage than Nick.
These conditions can be met in other ways.
Prone is already an option in the Open Hand monk and advantage is available through Darkness in the Shadow monk.
These other Weapon Mastery options can be available through the feat for other monks who don't want to multi-class and gives that feat some value to the monk class, but we don't need multiple forms of advantage.
Sap synergies well with Deflect Attacks, as getting hit once per turn will be far less than half the damage of getting hit twice.
My build concept uses Darkness. Opponents in that Darkness already attack the monk with disadvantage.
Push with the great club if your allies have zones up.
Definitely an option and alternative use for the feat but again built into Open Hand monks already plus it's situational.
The extra attack isn't situational. You've easily convinced me it's not overpowered, though. ;-)
Slow makes it even easier to kite people with javelin.
That looks like an overkill approach given the monk's inherent movement bonuses. If I am already giving up the bonus action attack from the unarmed strike for ranged kiting I can dash as a bonus action already with Step of the Wind.
Nick is only a little extra damage. Maybe if you got some bonus, like dip for hunters mark and a magic dagger.
It's a bonus that starts at d6 and ends at d12, and getting back to Darkness can typically be easily done with advantage already.
Nick has the same advantage here with monk's that it does with rogues in that it allows for that attack and keeps the bonus action available for other options that a monk can frequently use.
The advantage of using the feat is building towards an ability score the monk uses and not delaying access to higher level abilities.
That's not over powered but it does have it's uses.
Vex is probably the worst, since less than half your attacks are with the weapon.
Probably. But that gets back to using Darkness anyway.
The only issue with Darkness on a Shadow monk is the possibility of negatively affecting allies.
Your post was good food for thought on alternative choices for the use of Weapon Mastery.