Utility of Bent's Rule - What can Bent's rule explain that other qualitative considerations cannot?
There is a reason for everything.


  Does Bent's rule have any utility?


YES! It wouldn't be there if there was not. But I will get back to this at the end of this post.

Let me go through the points raised by your teacher first:


  
  Coulombic considerations can be used to rationalize bond angles, strengths, and lengths without the use of Bent's rule.
  


This might be true to some extent, but that only means that coulombic considerations give the same result as a complete orbital analysis. Boltzmann, Planck, Einstein, Schrödinger, Pauling, etc. worked the field for a reason - classical mechanics cannot be applied to molecular systems. Any theory that simplifies it beyond quantum mechanics, is a model to understand or comprehend - not a model to predict.


Orbitals are not real so why invoke orbitals when simpler Coulombic considerations are available? (Occam's razor)


Occam's razor would suggest picking molecular orbital or valence bond theory, as it states that from competing hypotheses the one with the fewest assumptions should be selected. Coulombic consideration is purely based on assumptions, it is a rather crude approximation to the correct picture.
The statement, that orbitals are not real is clearly wrong. There are many books that explain how one can comprehend this. Teaching that orbitals are not real is, with all due respect, just plain wrong.


The Coulombic effect is also real as opposed to orbitals.


The coulombic effect is not more real than orbitals. Orbitals should be used to explain the coulombic effect.



But let's talk about Bent's rule. The IUPAC goldbook states:


  In a molecule, smaller bond angles are formed between electronegative ligands since the central atom, to which the ligands are attached, tends to direct bonding hybrid orbitals of greater p character towards its more electronegative substituents.


It is important to know, that Bent's rule is, like Lewis rule of two (or eight, known as octet rule), a rule (not a law) based on observation. Hence its character is more or less one of a guideline. The key paper was published in Henry A. Bent, Chem. Rev., 1961, 61 (3), 275–311.

Bent's rule originates from valence bond theory (VB). Early forms of VB theory proposed that a molecule can entirely be described by a set (linear combination) of resonance Lewis structures, i.e. every bonding orbital is a directed localised orbital, which will be occupied by two electrons and the same applies to lone pairs. The advantage of this approach is that it is easily comprehensible. The disadvantage is, that it is quite inflexible. The reason for this was the use of either pure $\ce{s}$ and $\ce{p}$ functions or fixed hybrid orbitals $\ce{sp^3,~sp^2,~sp}$, which only allowed a subset of angles. A direct quote from Bent's original paper states this as follows:


  For atoms that satisfy the octet rule, and it is with such atoms (e.g., carbon, nitrogen, oxygen, fluorine) that this review is primarily concerned, the four valence-shell orbitals of an atom may be thought of as compounded from one spherically symmetrical $\ce{2s}$ atomic orbital and three mutually perpendicular dumbbell-shaped $\ce{2p}$ orbitals, designated $\ce{2p_$x$, 2p_$y$, 2p_$z$}$, ($x, y, z$ indicate mutually orthogonal axes). The $\ce{2s}$ orbital is lower in energy than the $\ce{2p}$ orbital; hence the ground state of a carbon atom, for example, is written $\ce{ls^2~2s^2~2p^2}$ (not $\ce{ls^2, 2s~2p^3}$. However, in a molecule such as methane, which is known to have the shape of a regular tetrahedron, the $\ce{2s}$ orbital of carbon may be considered to mix (or “hybridize”) with the three $\ce{2p}$ orbitals to form four equivalent orbitals that point to the corners of a regular tetrahedron and make angles with each other of $109^\circ28’$. These are called hybrid orbitals. Each contains one-quarter $\ce{s}$ character and three-quarters $\ce{p}$ character, abbreviated $\ce{s^{$1/4$}p^{$3/4$}}$ or $\ce{sp^3}$, or $\ce{te}$ (for tetrahedral) (223). In ethylene, on the other hand, the $\ce{2s}$ orbital of carbon may be considered to mix with but two of the three $\ce{2p}$ orbitals to form three equivalent hybrid orbitals that lie in a plane and make angles with each other of $120^\circ$; these three hybrids, which contain 33 per cent $\ce{s}$ character, are abbreviated $\ce{s^{$l/2$}p^{$2/3$}}$ or $\ce{sp^2}$, or $\ce{tr}$ (for trigonal); perpendicular to the plane of these three orbitals is the axis of cylindrical symmetry of the remaining pure $\ce{2p}$ orbital. The geometry of acetylene is often explained by assuming that the carbon $\ce{2s}$ orbital mixes with but one of the three $\ce{2p}$ orbitals, forming two $\ce{di}$ (for digonal) hybrids that point in diametrically opposed directions; these are abbreviated $\ce{s^{$1/2$}p^{$1/2$}}$ or $\ce{sp}$.


The original approach allowed already good results. However, Bent already draw the conclusion, that hybridisation is connected to molecular geometry:


  Molecular geometry provides then a direct clue to interorbital angles and
  (table 1) the distribution of $\ce{s}$ character. For example, from the bond angles in ammonia ($106^\circ46'$) (284) and water ($104^\circ27'$) (134, 284), it is inferred that the nitrogen atom of $\ce{NH}$, devotes slightly more $\ce{s}$ character to its bonding orbitals than does the oxygen atom of $\ce{H2O}$.


In conclusion to this (unfortenuately I ran out of time, so this may be extended soon) he suggested to use fractional linear combinations of hybrid orbitals to add more flexibility to the VB approach. Therefore Bent's rule is an extension to VB theory, that refers to the linear combination of atomic orbitals, that eventually form hybrid orbitals.

The formulated rule was later merely a result of the approach to flexibilize this theory. It is therefore a very important part of theoretical chemistry.

Of course there are now many more modifications to VB theory, and it is nowadays considered complementary to molecular orbital theory, but this would certainly exceed the scope of this question.