I would say the answer would most obviously be the money. They could have had the fame of saying they were there when the family was shot, but it seems like they would have to live in fear of being exposed in case the true story emerged since they would be lying and making details up from scratch. So it really wasn't worth the trouble.
As far as Botkin's children being able to ID him, yes that may seem true, but since they were either 'wrong' or lying about AA, how can we ever be sure they wouldn't have done the same thing for a father claimaint IF there were any money involved (which there wasn't, I assume if the good Dr. had any money they already got to it as his heirs)
Another thing to consider is that others who knew them well and had escaped, like Buxhoevedon, Vyrubova, Gibbes and Gilliard would also be able to expose them. Or even their own familes, if they were available. Which brings me back to AA. FS was a "Jane Doe" (Anna Anderson is the European name for Jane Doe or so I've heard) who easily claimed a new identity, and her family obviously was not anxious to claim her (probably considered her a constant problem and were glad to have others care for her) so she was able to play along more than people who could have been easily exposed.