I think we're imposing our own current thoughts re: "deprivation" on this topic. These days a 14 year old often feels "deprived" if he or she can't have multiple body piercings, wear hooker apparel and spend the entire weekend at the multi-plex eating junk food and terrorizing the average citizen.
The children of Nicholas and Alexandra were brought up in a rarefied atmosphere. What else would you expect of royal children? Even Princes William and Harry, for all of their mother's efforts, could not escape that situation . . . nor should they. If you have the means to keep your children sequestered from the hardscrabble ugliness of the world, of course you will do so.
Nicholas and Alexandra did not, however, want their offspring to be spoiled. And as the children grew older, they participated in experiences which encouraged them to be aware of others less fortunate.
Much has been written of Alexandra wanting to keep her daughters to herself, and therefore denying them relationships with others their own age. I do not deny there is probably some truth in this. However, I think the actual situation was a bit more complicated. Just who would OTMA socialize with? Aunt Xenia's children were primarily boys and, it appears, a fairly pugnacious group. Irina? Perhaps. But considering what happened later--that her parents gave permission for her to marry Felix--well, does that possibly indicate a preference for the accumulation of wealth over emotional stability? From all that I've read, this would not have been the case with Nicholas and Alexandra. Alexandra loved Nicholas for who he was; she had refused her English cousin, who was in line for the British throne, because she could not love him. Nicholas married Alexandra not because of her wealth--she was, by all accounts, a relatively impoverished and minor princess--but because he loved her profoundly. Parents with values such as these would be inclined to want the same for their children.
Olga and Tatiana had each other for companionship, but apparently very few others; the only friend I can think of at the moment is Rita Khitrovo. (Though of course we are not privvy to every bit of information re: this or any other area.) But again, given Alexandra's feeling of antipathy--in many ways justified--toward the "bright young things" of St. P society, it's no wonder the girls had limited social contacts. Still, they managed to attend dances, flirt, and enjoy themselves . . . we have contemporary accounts of this. I think that, given their "exalted" position, the two eldest girls at least did enjoy themselves and--had war not intervened--gradual freedoms would have been granted the youngest two. This is typical of the family dynamic; older children usually “blaze the trail” for those who follow.
Deprivation is a tricky topic. My sense of deprived teenagers might not be yours, and vice-versa. In a sense each one of us has had some sort of deprivation, whether it be lack of financial resources, an abundance of financial resources, permissive parents, strict parents, too little socializing, too much socializing, etc. In this life, no one can have it all. Parents can only do their best, and if they are empathetic and ethical, that "best" may be quite satisfactory.