When conducting a hypothesis test for a sample mean, we reject the null hypothesis when the test statistic is less than the critical value, or the P-value is less than the rejection region. Why is that when testing for a slope, (as stated above)we fail to reject the null hypothesis when the test statistic is less than the critical value?
We do not do as you say in the case of the sample mean. In that case, the test statistic is $t=\frac{\bar x-\bar x_0}{s_e}$, and $t$ is large in absolute value when the sample mean $\bar x$ is far from the null value $\bar x_0$. The situation is exactly analogous with the case of linear regression.
You didn't cite the text you've quoted, but it is contains a blatant error about one of the most fundamental aspects of hypothesis testing. We can never "conclude $H_0$", because the null hypothesis is that a parameter has some specific value, something that is impossible to conclude for a continuous quantity. I suggest you stop using this reference. The correct conclusion is that we do not reject the null hypothesis, which is entirely different than accepting it. In effect, we do nothing.