**Intro**

I'm the original asker of this question (9 months ago); thanks to the comments and answers I've gotten here, I think I've pieced together an answer that I'm happy with.

Short forms used in this answer: 

- CoLM = Conservation of Linear Momentum
- CoAM = Conservation of Angular Momentum
- KEB = the Kinetic Energy Balance
- CoE = Conservation of Energy

**Answer**

The equation derived in the question is actually KEB.  KEB doesn't provide as much information as CoLM does, but because of the symmetries of some problems, specific results are easiest to extract using KEB.  Full CoE, however, is a different equation, and does incorporate new information: it accounts for the fact that work, heat, and internal energy can inter-convert, connecting the motion problem to internal energy and heat transfer.  If the force laws are all known, then CoLM is all that is required to determine the motion, but CoE provides information about the sources/sinks in the of heat and internal energy - information which can't be extracted from CoLM. 

To address a few of the specific issues raised in the question:

- CoLM contains exactly as much information as F = ma does
- KEB is not a new constraint; it's sort of redundant on CoLM (in the sense that the two can never be incompatible), but [in two or more dimensions] it contains less information than CoLM and therefore can't be used in its place. In many cases, the symmetries of the problem are such that KEB alone can be used to extract useful results.  In those cases, the discarded information (which was in CoLM but isn't in KEB) pertains to aspects of the problem which aren't of physical interest.
- CoE is not redundant on CoLM; a new constraint has been introduced, but so have new variables (heat transfer and changes in internal energy)
- The 1-D elsatic collision problem actually ties everything together quite well.  Firstly, when solving with CoLM, adding KEB doesn't actually impose new constrains - as mentioned by steveOw, the new information comes from an assumption that the kinetic energy lost during the compression stage of the collision is equal to the kinetic energy gained during the extension stage of the collision, i.e., that the loss to heat/internal energy is zero, or equivalently the force at a given compression level is the same during both compression and extension.  This assumption is applied in KEB by setting the source/sink term to zero, but it could have also been substituted in CoLM (without KEB) to derive the same result (although the integration required would essentially re-derive KEB).
  
  Secondly, this is an example of how KEB can leverage symmetries of the problem to make it easier to extract specific results.  The symmetry here is the fact that the force at any point during compression is the same as the force at the same point during expansion. Per KEB, the full force law is actually somewhat irrelevant so long as it respects this symmetry - it doesn't impact the final speeds, at least. A full force law would be required to generate a full result, however; we know the velocities of the two objects, but we don't know how long they spent interacting, so we don't know what the starting point for the post-interaction paths was and thus don't know either object's position.  A full solution would require a full force law (and could also admit a force law without the compression/expansion symmetry), and the only equation used to fully solve the motion would be CoLM.

**Tangential Extra: CoLM and CoAM**

A related issue, which was touched on in the comments by David Z, is redundancy of CoLM and CoAM, and how this connects to the "conservation of constraints" idea of this question. CoLM and CoAM are kind of redundant, but kind of not, depending on whether you take the real-life perspective (that you can't have a force or moment which acts at a point) or the simplified perspective (that distributed loads can be simplified as a resultant point force and resultant point moment).  In the real-life perspective, there is a one-to-one correspondence between forces and moments - every moment is the result of a force and a moment arm - and CoLM and CoAM are redundant.  Continuum mechanics takes the real-life perspective (all forces are resolved as stresses - no point forces or moments), and so in continuum mechanics only CoLM is solved (CoAM would provide no extra information).  In the simplified perspective, the one-to-one force-moment correspondence is lost - there are point moments which don't have an associated force and moment arm - and CoLM and CoAM become distinct conditions.  Statics usually involves the simplified perspective, so in statics CoLM and CoAM (or, as they're more often applied, Fnet = 0 and Tau_net = 0) are distinct conditions.