Well, it's been an entire day, and no one has given an authoritative answer; I'll throw in my guess as to why the people designing DSA made the choices they did.
With DSA, there are three operations that are relevent to this discussion:
A: do precomputation of a signature (without seeing the message being signed)
B: given a precomputed signature and a message, generate the actual signature
C: verify a signature.
Standard DSA and your variant DSA perform exactly the same operations, except that standard DSA computes a modular inverse during steps A and C, while your variant computes it at step B.
It is true that standard DSA does two modular inverse, while your variant does only one; however, that may not be the only factor.
If we look at step B, we see that the operations involved at quite cheap (a single modular multiplication and addition); if we were to include a modular multiplication operation there, we would increase the cost of that operation by a large percentage. In contrast, steps A and C are already expensive (involving a modular exponentiation or point multiplication); including a modular inverse only slows down those operations by a small percentage.
That is, DSA has the property that, with precomputation, generating a signature is extremely fast. Your variant significantly reduces this advantage; the DSA designers designed to keep this advantage, even at the cost of increasing the total computation required.
Now, you may disagree with the above logic; in fact, people have proposed a DSA-variant exactly like what you suggest.