It is interesting to see this focus on the Moves tests in the States. While we do have skills tests (sort of/kind of moves in the field, but, much much much less involved) - they do not link in any way to freeskate levels, either in test or competitive track. You test freeskate, and that's that ... and if you are not in the test track, there is no requirement for stroking or other such in the test at all; in fact, they removed the requirement for minimum Skating Skills scores for competitive tests this year (although minimum PCS does apply, which does include skating skills, of course).
However, yes, my kids do have specialist coaches in what you would term Moves in the Field; we have a specific coach working on footwork, turns, edges, transitions, etc. Reasoning being that it is a key component of skating, and a primary coach "farming this out" to an expert in the area gives a benefit to the skater while freeing up time for the freeskate coach.