What is the purpose of the Maxwell Stress Tensor? In the calculation of the forces acting on a charge/current distribution, one arrives at the Maxwell stress tensor:
$$\sigma_{ij}=\epsilon_0 E_iE_j + \frac{1}{\mu_0} B_iB_j -\frac{1}{2}\left(\epsilon_0E^2+\frac{1}{\mu_0}B^2\right)$$
In the case of electrostatics, this element of the stress tensor denotes the  electromagnetic pressure acting in the $i$ direction with respect to a differential area element with its normal pointing in the $j$ direction. Equivalently, we can replace "electromagnetic pressure" with "electromagnetic momentum flux density" in order to "make sense". With this mathematical construction, assuming a static configuration, the total force acting on a bounded charge distribution $E$ is given by
$$(\mathbf{F})_i=\oint_{\partial E} \sum_{j}\sigma_{ij} da_j $$
Where $da_j$ is the area element pointing in the $j$ direction (e.g. $da_{3}=da_z=dxdy$).
What I would like to know is, what is the advantage of introducing such an object? I have yet to see a problem where this has any real utility. Sure, we can now relate the net force on a charge distribution to the E&M fields on the surface, but are there any problems where that is really better than just straight up calculating it? In an experiment, does one ever really measure the E&M fields on the boundary of an apparatus to calculate the net force?
 A: There are two big advantages to having a stress tensor calculated (and I'm sure there are others):
1) If rather than just the forces, you want to calculate strains and shears on an object that has physical extent.  then, you are interested in the off-diagonal terms of $\sigma_{ij}$, and these don't naturally pop out of a simple force description
2) General relativity is phrased in terms of the stress-energy tensor, the four-dimensional generalization of the stress tensor, so if you want to understand the matter terms in general relativity, you should definitely understand the stress tensor.
A: 
but are there any problems where that is really better than just straight up calculating it?

In the first place, it allows us to formulate general theorem of conservation of momentum in macroscopic electromagnetic theory. Suppose some amount matter is enclosed inside an imaginary surface $\Sigma$ in vacuum, no matter is present on the boundary itself but field may be non-zero anywhere. In a simplified terms, the theorem is
rate of change of total momentum in a region of space bounded by imaginary surface $\Sigma$ in vacuum = surface integral of Maxwell tensor $\sigma$ over $\Sigma$
or formally
$$
\frac{d}{dt}\bigg(\mathbf P_{matter} + \mathbf P_{field} \bigg) = \oint_{\Sigma}d\Sigma_{i} \sigma_{ij}.
$$

but are there any problems where that is really better than just straight up calculating it?

The Maxwell tensor is also utilizable in calculation of total EM force on a solid object, for example the force on electrically polarized body in external electric field or force on a magnet near metal body or another magnet. Such calculations can be done with pencil and paper for highly symmetrical systems like dielectric cylinder in charged parallel-plate capacitor or two magnet cylinders facing each other with their bases.
