In response to comments, it seems that the question is more along the lines of:
"what are some applications where computing summary statistics associated with a specific MSA result could be useful, even if we can't return the sequences?"
I have 2 responses to this:
in keeping with my original answer, I would say that it would still be useful to obtain some measure of identity, similarity, or pairwise distances between sequences (or any summary statistic about the relationship of the sequences). There are lots of applications for this level of data. I don't know whether this is something that can also be extracted from the computation, I'm a little unclear on what exactly the proposed computation is doing if it's getting a cost but not sequences.
Michael says below (and I agree) that an alignment cost is likely to strongly depend on particularities of the underlying alignment model as well as the data, and things get nasty when you try to account for gaps or weirdness. Nonetheless, it's possible that a cost could be used as a sort of guide heuristic for a separate follow-up alignment procedure, to indicate a portion of parameter space which is likely to yield good results. I don't know what this looks like, but it could be simple as resource provisioning for the computation (e.g. CPUs/RAM/disk). Alternately, maybe the cost result can be a little richer and actually seed some parameters in the next computation.
Again, without more detail I'm not sure what this looks like (and I'm not an alignment expert so my suggestions wouldn't be very good anyways).
- The simplest application might be to come up with a quick "check" of the sequences to ask whether they are alignable in the first place. Like, as a QC step (did you forget to reverse complement?) maybe. (this assumes that the alignment in question is pretty onerous and you don't want to have to try several times.)
In summary, a lot depends on how you get that cost and what else you could get in addition to the cost. I wonder if rather than focusing on cost you can output some other richer summary of the alignment (such as e.g. sequence similarities, or gapping stats, etc.).
The short answer is "yes".
Quite frequently we are interested in measuring "identity" or "similarity" of two sequence strings. As one totally arbitrary example, in metagenomics, quite frequently two rDNA amplicons that show 97% identity are considered to be in the same species, 95% same genus, 98% same strain, etc. So these measures are frequently useful as heuristics.
Identity itself is just a length-normalized edit distance across a sequence, after all.
But more generally, such measures of identity or similarity can themselves be understood as cost measures, in the sense that e.g. a dynamic programming alignment algorithm is trying to find an optimal set of matches between two sequences, which necessarily will maximize identity. Generally what it is actually maximizing is similarity (according to some cost matrix such as e.g. BLOSUM or Jukes-Cantor or whatever).