Singleton removal is quite a common procedure in the literature, but it's a complex and difficult topic. See
this paper for an in-depth technical analysis on the consequences of different parameter choices in these sorts of analyses. Singleton removal has a large effect on alpha diversity estimates -- but whether the estimates are more
accurate before or after singleton removal depends a lot on your assumptions about what those singletons represent.
In general, I think our ability to make comparative assessments of diversity within a study using these methods is still much greater than our ability to make precise absolute statements about the starting sample.