Why don't people trade significance level for power? 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?
 A: This is more of an extended comment than an answer.  One interesting perspective can be found in this blog post, a short citation:

... contends the word [significance] carried much less weight in the
  late 19th century, when it meant only that the result showed, or
  signified, something. Then, in the 20th century, significance began to
  gather the connotation it carries today, of not only signifying
  something but signifying something of importance. ...

If this is correct then Fisher can have meant with significant rather something like worthy of taking a note (mental or in lab notebook), worthy for further investigation or replication. 
This psyarxiv paper proposing to reduce standard significance level (in psychology research) from 0.05 to 0.005 is further evidence that many see (rightly ...) that 0.05 is already a rather weak requirement. 
A: 
Why are experiments with moderately small $\alpha$ and very high power so rare?

This is all a bit relative, but one could certainly argue that the significance level $\alpha = 0.05$ is already weak, and already constitutes a sacrifice made for higher power (e.g., relative to the significance level $\alpha = 0.01$ or other lower significance levels).  While opinions on this will differ, my own view is that this is already a very weak significance level, so choosing it at all is already a trade-off to get higher power.

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.

I can see why you might think this, but it is not really true.  In classical hypothesis testing there is quite a complex and subtle interaction in these things.  Remember that both the p-value and the power pertain to probabilities that condition on the true state of the hypotheses (the p-value conditions on the null, and the power conditions on the alternative).  When you get your result from the data, you make an inference about the hypotheses, but you still don't know their true state.  Thus, it is not really legitimate to say that you can completely ignore the "other half" of the test.  Regardless of whether the result is statistically significant or not, the interpretation of that result is made holistically, with respect to all the properties of the test.
It is also worth noting that, for a fixed model and test, and a fixed sample size, the power function is a function of the chosen significance level.  The chosen significance level determines the rejection region, which directly affects the power of the test.  So again, there is a relationship between these things, and you cannot ignore "one half" of the properties of the test.
Finally, it is also important to note that practitioners conducting a classical statistical tests will often just report the p-value of the test and leave it to the reader to choose their own significance level if a binary decision is required.  (That is my preferred approach unless there is a specific need to make an immediate binary conclusion.)  Modern statistical literature cautions strongly against reducing reported outcomes of hypothesis tests to a binary without also giving the underlying p-value.  So in many practical cases, the significance level is not chosen prior to the analysis, and might not be chosen by the analyst conducting the test at all.
A: Because type II errors are considered to be less of a problem than type I errors. Type I errors have greater implications for future research. Moreover, most of the time, experiments with high power are much more expensive.
But of course you can also question both the whole NHST framework and the way it is frequently misused by unaware researchers...
A: In a hypothesis test the null hypothesis and the alternative are not symmetric. Hypothesis test logic is connected to falsification logic. The idea is that one wants to be able to make a strong statement of having "statistically disproved" the null hypothesis in case of a significant result. Surely this requires a small significance level; 0.05 seems rather high in this respect, apart from the fact that in the meantime fishing for significance by running lots of tests and selectively reporting significant ones has made a mockery of the idea to "statistically disprove" a null hypothesis (which is one reason why some people advertise lowering the threshold to 0.005, although this won't stamp out selective reporting).
In fact if you run a test with significance level 0.2 and power 0.95 for a specific alternative, apart from subtleties with composite hypotheses, this means that rejecting the null hypothesis means hardly anything, as this will happen all the time (OK, 1/5 of times), whereas not rejecting is clear evidence against the alternative. Basically you run the test the other way round.
A: 
Why are experiments with moderately small α and very high power so rare?

High power is no problem, but the culprit is with the moderately small $\alpha$.
Experiments with only moderately small $\alpha$ are not precise and are prone to observe apparent effects that are not reality and only arrise due to imprecision and noise.
These imprecise, moderate $\alpha$ experiments are rare because the scientific community wants to read about results that are obtained with precision. To perform experiment with some small $\alpha$ is a scientific standard of performing an experiment that is sufficiently precise.
High power can not cure an experiment that is lacking precision. The high power does not change the uncertainty about the results.

so that you can confidentially accept the null

This is very similar to a test of equivalence in which you test the absence of a particular effect.
You do not really 'accept the null' only because you did not measure an effect. But you can reject some sort of the non-null, that the effect size is beyond some level, and accept the alternative that the effect (if it exists) must be small below some level.
