This is not a thread for discussing whether Anna Anderson (or anyone else) was really Anastasia. Rather, it is meant for a discussion about whether it would have made any real difference in the course of later events if she (or any of the children) did survive.
If I look at the fates of people who were incontestably close relatives of the last tsar, they fared rather poorly as a lot. Other than the Dowager Empress, most of them went their separate ways and forged whatever lives they could. Of those who did not trade on their names to marry heirs and heiresses, Xenia had perhaps the best of it, living in a grace-and-favor house on the charity of the British Royal family, who paraded around in jewels once owned by the Romanovs. Marie Pavlovna the Younger had to work for a living, retrieving material from storerooms where she stumbled upon extra bolts of fabrics that once graced her family's apartments at the Catherine Palace. One Grand Duke (whose name escapes me) was reported to be a chauffeur in Paris. And Olga Alexandrovna, the last and closest surviving relative of the last tsar, died in poverty over a barbershop, cared for by neither friend nor family.
The Romanovs as a ruling clan were not highly regarded -- inside or outside of Russia -- in the years immediately preceding and following the revolution. Most of the wealthy people who took an interest in their fates viewed it as an opportunity to pick off the spoils of imperial wealth -- Marjorie Post, Armand Hammer, Malcolm Forbes, even Queen Mary. Other than to a few snobs who were desperate to acquire a royal pedigree (even a deposed one), the Romanovs were mostly curiosities on the world scene, relished as much for the collapse of their fortunes as for anything else.
By those standards, Anna Anderson's fate of having an eccentric but affluent professor jump to her bark right up to the end was really not too bad. In fact, I can find nothing in the treatment accorded other members of the tsar's family to indicate she could have expected any better.
If she wanted international attention, she got more than either Xenia or Olga. If she wanted physical and financial security, she got more than Olga and many others. If she wanted to be recognized as who she said she was, she fared well enough in having some people recognize her, given the difficult case she presented to her supporters.
If it had been incontestably proven that Anna Anderson was Anastasia, what difference would it have made to the course of subsequent history?