Dasha, I agree, parents who expect others to fawn all over their children are doing their children a profound disservice. And what a ego trip it is, since these types of parents see their children only as extensions of their own psyches, and not as personalities-in-training who need guidance and guidelines!
But I was trying to put across that children who are in the three to four year old age bracket
are at a stage when they are apt to strut a bit. Or, as I remember from a long-ago class plus my own subsequent experiences, "the terrible twos" is a bit of a misnomer . . . it's actually right about three years of age when things can get hectic!
Understandably Nicholas and Alexandra were proud of their son. Not to mention protective, for reasons that we all know. But from what I've read, I think they were doing their utmost to mold him into a well-behaved tsarevich. Obviously Alexandra indulged him. And we've read accounts of Alexei's less-than-commendable behavior, plus others besides myself have mentioned that irrefutable bit of film in which he smacks a cadet.
It's a rare child, however, who can bear 24/7 scrutiny of his behavior. I remember a boy from my childhood--also the family's only son--who was great fun as a child, then around the age of 10 became someone you definitely wanted to avoid. Foul language, foul behavior--especially toward us girls--etc. Well, guess what. He grew up into the most approachable, congenial member of the family . . . and the only one I've kept in touch with. When I think back to Mike's former behavior, though, it makes it very easy to visualize the naughty to downright rotten behavior "Alexei the Terrible" could exhibit. Conversely, I also can visualize the kinder side of Alexei which so many memoirs have described.
As for those less-than-wonderful episodes, being a boy is part of it, if I may be forgiven that sexist statement!

Then there's the matter of approaching adolescence. And, of course, being told that you are the tsarevich. A child can mistranslate that message--probably meant to convey a sense of responsibility--into one of self-importance. Just think of any child brought up amongst a household of servants . . . I can think of a number of real-life situations, but to be diplomatic, consider the bumptious little Mary of
The Secret Garden.

Figure in the hot-house atmosphere meant to protect Alexei from injuring himself, and it all adds up to an interesting and in some ways unique case history, worthy of a child psychology thesis!