As a convention, we have a lot of studies whose significance level is $0.05$ and a power of $0.8$. However, it is extremely rare to find a study whose $\alpha = 0.2$ with a power of $0.95$.
From my understanding, after an experiment has been conducted, the significance level doesn't matter at all if the result is non-significant, because in this case, we are considering whether it makes sense to accept the null, and all we care about is the power. Similarly, if the result is significant, then the significance level becomes your evidence, while the power of the test makes absolutely zero difference. (By "doesn't matter", I mean "doesn't for the purpose of this experiment". Both significance level and power should be important for meta-studies, so please report both in your publication!)
If I'm correct, then the null and the alternative are to some extent symmetrical: the null hypothesis doesn't inherently require more protection. If you want to prove the alternative, say "this new drug has an effect on the patients", then use a very small $\alpha$ and moderately high power. On the other hand, when you want to prove the null, for example in a normality test, then you should choose a moderately small $\alpha$ and very high power, so that you can confidentially accept the null.
Why are experiments with moderately small $\alpha$ and very high power so rare?