Let's look at how a quadrotor flies, then apply that to a trirotor.
Let's assume that we want to remain in a stationary hover position. To do that, you need to balance all the forces: thrust from the propellers vs. gravity, and the torques of each motor.
Each motor produces both thrust and torque according to the equations:
$$
T = K_T\rho n^2 D^4
$$
$$
Q = K_Q\rho n^2 D^5
$$
Where $T$ is thrust, $Q$ is torque, $K_T$ and $K_Q$ are system dependent constants, $\rho$ is the air density, $n$ is rotor speed, and $D$ is rotor diameter.
If you increase thrust then you increase torque, and vice versa. A quadrotor remains stationary by balancing all the forces. This is possible because the quadrotor is symmetrical: two motors spin clockwise, and two motors spin anti clockwise. If all the motors rotate at the same speed then the torques balance, and the thrust balances.
The only question then is what speed? The four rotors need to spin fast enough to collectively generate enough lift to remain in a stationary hover.
What about a trirotor?
Intuitively, the easy base case is when the arms holding the motors are all the same length (such that you can ignore the effects of the motor's displacement from the center of mass). In this case you must set the force of each motor to be equal (to hover without falling) then the torques will be unbalanced (2 in one direction, 1 in the other). The result is a spinning trirotor.
The slightly more difficult case is when the rotor arms are not the same length. To solve that, let's solve the general case. The equation for torque is:
$$
\tau = rFsin(\theta)
$$
$sin(\theta)$ is 1 (at a right angle), so we can ignore it and rearrange our torque equation as follows:
$$
F = \frac{\tau}{r}
$$
Let's make a trirotor and label it as follows:
Then, we can balance all the torques:
$$
\frac{\tau_A}{r_A} + \frac{\tau_B}{r_B} + \frac{\tau_C}{r_C} = 0
$$
Substitute in the motor equation for torque from above:
$$
\frac{ K_Q\rho n_A^2 D^5}{r_A} + \frac{K_Q\rho n_B^2 D^5}{r_B} + \frac{K_Q\rho n_C^2 D^5}{r_C} = 0
$$
And get rid of the common constants:
$$
\frac{n_A^2}{r_A} + \frac{n_B^2}{r_B} + \frac{n_C^2}{r_C} = 0
$$
To solve this equation, we need to make a system of equations with the thrust for each motor:
$$
T_A + T_B + T_C = mg
$$
$$
K_T\rho n_A^2 D^4 + K_T\rho n_B^2 D^4 + K_T\rho n_C^2 D^4 = mg
$$
$$
n_A^2 + n_B^2 + n_C^2 = \frac{mg}{K_T\rho D^4} = C
$$
Where C is some constant. We don't really care what it is as long as it's non-zero (if it's zero then we don't need the motors to do anything).
Now, make our system of equations and solve them:
$$
\frac{n_A^2}{r_A} + \frac{n_B^2}{r_B} + \frac{n_C^2}{r_C} = 0
$$
$$
n_A^2 + n_B^2 + n_C^2 = C
$$
Right away we see that there is no solution except when $C=0$.
Thus, a trirotor needs the servo to rotate at least one motor in order to generate more torque without generating more thrust (in the z direction).
