How does the block discovery rate of a mining rig relate to the overall computing power of the network? Is it correct to say that, since the difficulty will keep the block generation rate at 6 per hour on average, that with one-sixth of the network computing power, I will get one block per hour, on average?

Put differently, can I express my expected average block discovery rate to always be 6 per hour multiplied by the fraction of total computing power I own?

I'm asking this because I keep getting confused by the profitability calculations that rest on the difficulty, which is a totally opaque value for me, even though I know in theory what it means. I would like to employ some measure that I can use more intuitively.
This is approximately correct. You will actually get a bit more than one block per hour because so long as hashing power is increasing, the difficulty will lag behind the actual computing power.