I tend to get confused by the issue, so I just worked it out on paper.
Not all of the Queen's first cousins became full time working royals. That would tell me that there is not precedent that a prince or princess must be given digs at KP and a job as a full time royal.
None of Charles first cousins are full time working royals. Again, this tells me there is not a precedent.
Just because you are a princess or prince, there is no precedent saying you must be given work as a full time royal. Two generations back.
There's only one generation of precedence actually.
George V had 5 children to survive to adulthood; Edward VIII, George VI, Mary, Henry, and George. Edward had no children, and the children of Mary were never royals.
George VI, of course, had two daughters, Elizabeth II and Margaret, who did royal duties for most of their lives.
Henry had two sons, and it seems like it was kind of expected that his elder son (William) would in time become a working royal, but he died rather young. Instead, the younger son (Richard) gave up his career as an architect to work for the family, so to speak, when William died.
Similarly, George had two sons and a daughter (Edward, Michael, and Alexandra). It was expected that Edward would work for the family, while Alexandra and Michael were expected to make their own way - Michael went in the military, while Alexandra studied to be (I believe) a nurse. Alexandra's private career got kiboshed (much like her cousin's) with the early death of George VI, and with a relatively small royal family she began doing full time royal duties.
I think it's kind of important to remember that in 1950 Queen Mary and 6 of her children and daughters-in-law were working royals, but by 1960 that had been reduced to just 1 child and 3 daughters-in-law, and the Queen was a woman with young children. Her cousins, those that ended up working full time, were essentially conscripted into it by the deaths of those before them.
Now, as for the descendants of George VI; Elizabeth of course became Queen, and Margaret was a full time royal for much of her life. Elizabeth's children have followed suit, but Margaret's fall into the same category as Mary's children - they're not royals, have never been, and therefore don't do royal duties.
In this generation, looking at the Queen's grandchildren, Anne's clearly follow the precedent of Mary and Margaret's - not royals, never have been. Charles' clearly fall into the precedent of being the future of the family. But Andrew's and Edward's are in an in between - they could end up with private careers, but they might get called up (like the Gloucesters and Kents did).
I actually tend to think that Edward has taken moves to make it unlikely that his kids will ever be seen as royals, and therefore will avoid that role altogether. And I honestly don't think Eugenie wants the role, I think she's happy to have a private career. Beatrice is different, and I think she's been put into an awkward position by life. In time there could very easily be a full time role for her - unless the plan is to drastically reduce the number of engagements once the Queen and her cousins pass, Beatrice could easily step into a void that will open up in a few years. But because William, Harry, and Kate aren't full time royals, and because the Gloucesters and Kents continue to work (to varying degrees), Beatrice is in a spot where she can't do more... but if the plan is for her to one day step into that role, then she'd also be silly to seriously pursue a career that she'll one day have to give up - I somehow doubt the Queen wants her granddaughter to have to make the same sacrifice that two of her cousins had to make.