Morgan and Elliot showed that an array's sensitivity to perturbation is directly related to the white noise gain of the array,
E. N. Gilbert and S. P. Morgan. Optimum design of directive antenna arrays subject to random variations. Bell System Technical Journal, 34(3):637
They showed that the most robust array has equal magnitude weight coefficients.
There are more than a few reasons to diagonally load a covariance matrix, like in Dominant Mode Rejection, but for the case of a perturbed array, on page the sensitivity increases as $|| \mathbf{w} ||^2$ increases.
If you refer to page 505 of :
H.L. Van Trees. Detection, Estimation, and Modulation Theory, vol 4: Optimum Array Processing. Wiley, 2004
It is shown that adding a quadratic inequality constraint, $\mathbf{w^H} \mathbf{w} \le T$ (i.e. bounding the coefficients), MVDR beamformer will have the form:
$$ \mathbf{w^H} =\frac{\mathbf{v^H} (\mathbf{S_x} + \lambda \mathbf{I})^{-1} }{\mathbf{v^H} (\mathbf{S_x} + \lambda \mathbf{I})^{-1} \mathbf{v} }$$
So, diagonal loading has the effect of reducing the spread in amplitude of the weights. A large of spread of magnitude weights is also a characteristic of having interferers in the main beam, and a quadratic constraint "protects the main response pattern"