First of all, the variance is not as high as you think. It doesn't take hundreds of attack rolls to start matching the predictions pretty well, it takes a few dozen.
Do remember that the average you may see after a few dozen rolls is not - "generally I do about average", but is instead likely to be some notable highs and lows - in terms of perception and actual effect in game, those are not the same thing.
I think character advancement also has an impact on the perception. You've typically got a dozen or so encounters before you level up, right? A few rolls per combat - so you only just get that few dozen rolls before the character,
changes, altering the statistics and performance again.
Second, even if the die rolls at the table don't exactly match the predictions, the predictions are still useful and important because they tells us the shape of what we should expect to see as the die rolls vary.
The statistics are good for design, QA, stress-testing, and setting some base expectations.
But I think many folks say, "the variances will be small, so we won't worry about them", but I think they are missing a major point. There's tens of thousands of players out there - which means the distribution will get thoroughly filled out. Yes, most will see average performance. But the tails of low and high performance will be filled out too. And human perception notes extremes more strongly than nominal behavior.
Thus, in some senses, what happens when things go wrong is
more important than what happens when things follow expectations.