It seems to me that my computer downloads/verifies blocks at the rate that is lower than the rate they are mined.
Seriously, do you think the cause of this situation is with Bitcoin (thus a majority of users would experience this), or with your particular computer?
(Hint: title should be "my computer is becoming a black hole")
I believe I would outpace the generation rate but advent of ASICs made the situation worse and now I have to switch to a more powerful computer.
The coming of ASICs, or the total hashing rate of the entire Bitcoin network for that matter, has absolutely NOTHING to do with this whatsoever.
What exactly do you mean with "outpace the generation rate"?

The generation rate is 1 block per 10 minutes,
by definition. Do you even understand how Bitcoin works?