We never accept the null hypothesis. We either reject it in favour of the alternate hypothesis or we fail to reject it. It sounds like we're splitting hairs, but the distinction is important.

When the p-value > 0.05, we do not reject the null hypothesisbecause there is not enough evidence to do so. It doesn't mean that the null hypothesis is true. This is something that is much harder to prove.

It's a similar situation in a criminal court. The judge decides if the accused is guilty or not guilty. The judge does not decide whether the accused is guilty or innocent, because innocence is much harder to prove and the onus is on the prosecution to demonstrate that the evidence is overwhelmingly pointing to the accused being guilty. Similarly, the onus is on the researcher to prove that the alternate hypothesis should be adopted instead of the null hypothesis.

Let's put it another way. Let's say that you have a sample of 100 observations, and a test you ran came up with a p-value of 0.051. Would you conclude that the null hypothesis is true? Maybe if you had collected a handful more observations and you re-ran the test, the p-value would have been < 0.05. So why would you jump to the conclusion that because something is not proven to be something other than 0, it HAS to be 0? That's a leap in logic.