Why MySQL does not provide stddev(distinct expression)? I ask the question because some other numerical aggregates, like sum, avg or count support the distinct modifier. And other databases like Oracle or MSSQL support distinct modifier for stddev.
Possible answer:  Standard Deviation is one of thousands of things that are so little used that it is not worth providing.

You can provide it, even in summary tables, by keeping

COUNT(*)
SUM(x)
SUM(x*x)


There are two stddev forumlas; they differ by only tiny amounts for large sets of data.  One of them can be reformulated using only those 3 aggregates.