excellent posts earlier by NextLevel and UpSideDownCarl ... my two cents ...
Is wrist an essential component of any table tennis stroke .. yes ... tabletennis is a very fast game so you will need to use your small muscles as much as possible the closer you are the table . Here is a golden rule of technique one of my coaches told me :
1. When you are inside the table ( flick + push ) , wrist is joint you should use for pivot ( I should have asked whether its true for FH pushes but I didn't
)
2. When you are playing close to the table , elbow is the pivot .
3. When you are playing away from the table shoulder becomes the pivot.
Again , this does not mean you are not going to use combinations or you can choose to use smaller pivots for consistency even when you are outside the zones marked. e.g. because of the speed of the ball a lot of techniques prefer to primarily use wrist on the BH side even when you are not inside the table but close enough.
Now the degree and how to engage wrists vary based on different factors. e.g. when you are developing your forehand loop , and you find the issue of mistiming mentioned above , the coaches tend to ask you to lock the wrist . Less number of moving joints = easier to have more consistent power and movement = more chances of making the shot. But this is true only during the development of the stroke , when your stroke is matured it will will include wrist . should include wrist unless you have presistent issues either physically or stroke wise.
The way it works for me , I just let it flow the way Carl mentioned , that is I keep the wrist loose enough so the kinetic chain mentioned by Nextlevel does not finish till it causes the wrist to complete its movement.
Now is the amount of wrist involved same in sidespin cross court loop in the short corner , a standard cross court loop, a forehand down the line and a fade ? of course not ! They are different strokes , its like comparing FH loop vs BH loop drive ...