My thoughts:
The SPORT of figure skating is organized for the purpose of demonstrating mastery of figure skating techniques and choosing who was best, second best, overall in each competition.
Part of the demonstration of technical mastery is shown by moving in aesthetically pleasing ways and in time with music, if possible reflecting the emotional content of the music.
Not all competitors will skate in ways that are pleasing to watch, and not all competitors will skate in ways that match and enhance the music. They're rewarded if they do, but it is possible to meet the requirements of competition and even to place relatively well at some levels if those aspects are the skater's weakest points.
It is also possible for skaters who are interested in creating an intentional coherent aesthetic/emotional experience for viewers to transcend the limitations of competition and create performances that could be considered "art" within the competitive context, but that is rare and not a required part of the sport.
The primary purpose of SHOW skating is to entertain audiences using skating skills/techniques and other performing arts skills and techniques.
The emotional context of a skating show performance may be just "This is fun!" or it might be more nuanced and/or serious. There are different ways to entertain audiences, and there is room to debate whether it is meaningful to make a distinction between art vs. entertainment. I'm inclined to be inclusive and say that if the goal of a performance is just to entertain and it succeeds in doing so, that is a valid artistic expression.
Some forms of skating shows (what we call professional Ice Theatre, and also some shows headlined by big competitive stars that aspire to do more than just showcase skills) tend to aim for performances that aim to create serious nuanced "art" in the same way that stage dance performances do. In fact, these groups often collaborate with choreographers from the world of artistic stage dance (ballet, modern/contemporary dance, etc.). And choreographers who started out as competitive skaters themselves may develop into artistic choreographers for the ice.
To me, the purpose of these shows is to create art. We can have different opinions about which ones are more artistic than others, which speak to us individually or not, same as we might have different opinions about which kinds of dance or other physical performance genres speak to us most. E.g., I would include circus performances, especially those using human movement, e.g., clowning, daredevil acrobatics, as comparable to dance in this way, although the specific emotions emphasized might be different.
I think these types of skating shows demonstrate that the techniques of figure skating can be used for the creation of art just as other types of movement performances off the ice do.
So if we ask "Is figure skating art?" my answer would be "It can be."
But in a competitive context, I think that any art that exists is an extra quality added to the main purpose of demonstrating technical mastery.
(But I did write a paper once comparing geometric motifs in Navojo sand paintings to Tibetan Sand Mandalas -- exploring art while sticking to my own lane. 
