How is formal analysis of the strength of a round achieved?
diehard is not the answer you're looking for. diehard is a randomness testing tool. It looks at a sample of ~10MB and checks if that sample is  independent and uniformly distributed (IID).  It uses simple statistics not implicit cryptographic experience.  The output is essentially a binary output in that if some p values = 0.00000 or 1.00000 then the sample is not sufficiently IID.  The sequence of p's that you'd get for a sample unequivocally known to be IID is  somewhat a matter of interpretation, feelings and guesswork.   dieharder, it's successor makes this binary choice much more explicit in outputting PASS, WEAK or FAILs.

You could easily produce a 10MB sample from the first round of a Feistel network such as DES. It would probably fail with a series of 0's after the decimal point.  As the round count increases, the p values would start to look reasonable, but not increase any further.  You would learn little of security from this exercise other than that the output is IID. So is 10MB of RC4 when analysed with diehard.  

Don't be surprised that key expansion inside some of these primitives is weak.  AES is weakish.  Key expansion is only a part of the entire primitive and their security comes from a sum of the parts.

PS. Colourful language is great, but in this particular instance categorising key expansion algorithms as ridiculous skews the question somewhat.  I think that analysts call them weak, rather than ridiculous.  "Ridiculous" would transfer all the security responsibilities to the main part of the primitive, and imply that any old expansion like ROT13 could be used.