The halving could come from an intended use in a Newton Raphson iteration of a square root refinement.

See for example https://math.mit.edu/~stevenj/18.335/newton-sqrt.pdf

The initial guess is the approximate square root, but it needs to be halved as part of the calculation.

It's probably not about avoiding extra instructions/performance, but making the range of the result more useful and avoiding overflow. Or in other words, the entire instruction may be useless if you don't do these things.

The halving and clamping is nothing particularly remarkable in the context of usefully using fixed point numbers (scaled integers) to avoid overflow. Reciprocal square root itself is a fundamental operation for DSP algorithms and of course computer graphics. This is a fairly generic instruction really, though FRSQRTE likely gets more real world use.