'As far as the Dowager was concerned she had the fortunate position of raising healthy children who required minimal personal attention, unlike her daughter-in-law who personally took on the role of motherhood. Her responsibilities towards Alexei took precedence over organizing grand balls and entertaining persons who grew accustomed to such Court amusements during Alexander III's reign.'
Surely organising balls was a matter for the household. Alexandra could be as much or as little involved as she wished. Here the Queen is well-known for her interest in the minutiae of state occasions (she always inspects the place settings for state banquets), but things would happen perfectly well without her doing anything. Realistically, all Alexandra needed to do was decide what outfit she was going to wear, perhaps cast an eye over the guest list, and turn up. Since the children had nannies she would not even need to find a babysitter, and, however obsessive she was over Alexei, she would only be away from him for a matter of hours. I'm guessing slightly here, but I would imagine that she and Nicholas would only need to turn up at a ball for a couple of hours at most, and could then disappear.
I don't think balls were an innovation of Alexander III's reign. The catalogue I have for the Hermitage Amsterdam exhibition 'At the Russian Court' (very well worth a visit if it's still on) has a good deal to say about balls and dinners under Nicholas I and Alexander II.
'Surely it was up to the Russian Empress to structure her time in her Court as she saw fit?'
Not entirely. There were things that were 'part of the job'. Just to draw a parallel, King George VI was terribly shy and, in particular, hated reviewing troops. However, it was part of his job as king so he did it - though there is a tale that when visiting the Eighth Army in North Africa in 1943 he had to be coaxed out of his tent for the purpose! One reason he is held in considerable affection is that he struggled to overcome his speech impediment and did things he'd have preferred not to do (such as being king in the first place!)
Ann