I believe you are asking "all moving vertical stabilizer" because FC-31's horizontal stabilizer is all moving.
No, it has nothing to do with low speed or landing on CV. J-20 designer 宋文骢 has explained why J-20 choose the full moving vertical stabilizer in his paper 一种小展弦比高升力飞机的气动布局研究.
The logic goes like this.
J-20 uses leading edge + canard to increase lift => high pressure between vertical control surface and aft fuselage => demand smaller vertical control surface. However a reduced vertical stabilizer + rudder design will reduce control efficiency. So the choice is all moving rudder to maintain its efficiency while reducing size.
The study that leads to above conclusion was based on comparison between an ideal tailless design base and J-20 at Mach 0.2 and AOA 24 degrees. You can see that this is a very low speed.
FC-31 on the other hand does not use complicated canard and leading edge vortex, therefor it choose stabilizer + rudder design which is easier in flight control design.
