"With these methods, you’re providing evidence against the hypothesis that nothing is happening at all."

Close, but not quite. If nothing is happening at all, there would be no need for a test. The null hypothesis is testing whether what you see happening is simply due to chance. When you run a test like a regression, you are measuring something that is really happening, but a p value less than, say, the standard .05, means that if you ran the same test 100 times against different data sets, you'd get similar results at least 95 times. The point is that you can make decisions based on such results. But if the p value is over .05, the observations could just be chance, and you'd be wise not to act on them.