If mining hardware was instead dedicated to generating new addresses; how long do you think would it take till someone stumbled on an existing address that had more BTC stored than what the person would have earned by mining?
Never. Even if the whole generating network was behind it, it would probably never even stumble upon a previously used address, much less one that is worth more than the number of blocks it could have generated instead. Generating blocks is trivial compared to this.
Their are 2^160 possible addresses. Lets say 2^32 (4 billion) people use Bitcoin and each generate 2^16 (65 thousand) address. That gives us 2^48 total addresses out of 2^160 possible. The probability of a generated address matching one of these is 1/(2^112). The probably of finding a block at 35 times the current difficulty is around 1/(2^64). Therefore, it would take 2^48 (281 trillion) times longer to find a previously used address. So if it takes you ten minutes to find a block, it will take over five billion years to find a used address.
Now keep in mind that we don't have 4 billion users, most users have far less then 65 thousand addresses, and the current difficulty is much lower then I used in these calculations. Mining sounds a lot more profitable.
NOTE: I assumed generating an address took equal time as generating a proof-of-work hash. However, I believe generating an address is actually slower since it involves both EC key generation and hashing.