Fooling arbitrary symmetric functions
Yes, a general solution to this problem has recently been given by
Parikshit Gopalan, Raghu Meka, Omer Reingold, and David Zuckerman,
see Pseudorandom Generators for Combinatorial Shapes.

That paper handles an even more general setting, where the generator outputs $n$ $\log m$-bit blocks, which are then fed to arbitrary boolean functions, whose $n$ outputs are then fed to a boolean symmetric function.

A variety of sub-cases were already known; see for example Pseudorandom Bit Generators That Fool Modular Sums, Bounded Independence Fools Halfspaces, and Pseudorandom Generators for Polynomial Threshold Functions. The first handles sums modulo $p$. The second and the third handle precisely the threshold tests you mention, however the error is not good enough to apply your reasoning to obtain a result for every symmetric function.