Do I have the right idea about statistical power? I've been struggling with an intuitive way to grasp power and hypothesis testing in general, and I'm wondering if this is the right idea:
Let's use the example of a random variable $ X $, following normal distribution $ N(\mu, \sigma) $, with unknown mean $ \mu $ and known variance $ \sigma $. We have a null hypothesis $ H_{0}: \mu = \mu_{0} $, and alternative $ H_{1}: \mu = \mu_{1} $. For the sake of power analysis, we assume that $ H_{0} $ is false, and $ \mu = \mu_{1} $. But, we don't know for sure if the null is false, and it doesn't really matter if it is anyway; we just want to see how well our hypothesis test can detect this difference $ \mu_{0} - \mu_{1} $, in the case that the null is false. 
Is this a good idea? It seems that, in hypothesis testing as a whole, there is a lot of "assuming [this thing] is true", not necessarily because we know it's true, but because we need to see what happens in the event that it is true. 
 A: This:

our goal for power analysis is to find the most powerful test given α

is not the usual goal of power analysis. Rather, the goal of power analysis is usually to either a) Find the sample size needed to have a reasonable chance of detecting a given effect or b) Find the chance of predicting a given effect given a sample size.  Usually we take $\alpha$ as fixed (at 0.05 or 0.01, most commonly), and we get a notion of the effect size that we want to be able to detect from the literature or theory.
We don't choose statistical methods based on which is most powerful but on which answers our research questions while not having any assumptions that are violated. 
A: The point of power calculations
The whole point of power calculations is that we do not yet exposed data to our question. This means we have no feeling for what it will tell us about our expected outcomes. We may not have the data or we if we do, we wish to be prudent in only applying the most useful tests in order to reduce the risk of type 1 errors (incorrectly affirming effects that don’t really exist).  This means we cannot at this stage formulate questions of the type ‘based on this data what is the probability our hypothesis is true’ precisely because we are not at the data analysis stage yet. As soon as we open the data box, we cannot push it back in so we want to optimise our analysis strategy first.
Is this a good idea?
Yes it certainly is, it derisks your analysis plan, ensuring it remains focused on realistic goals and minimises the risk of leading you down false alleyways that turn out to be dead ends. 
If you have ill formed concepts or expectations about the data then power calculations will not be based on your knowledge of the likely distribution of your data, but rather based on the application needs. If you know that a certain effect size ($ \mu_{0} - \mu_{1} $,) is required to be practically meaningful (too many scientific publications conflate statistical and practical significance,  result can be statistically significant but of no practical value), then you would power based on this required benchmark.
Powering allows you to choose a level of risk ($\beta$) that your are willing to take that you will accept a false effect. Often the default is 80% but realistically this should be based on a risk-benefit analysis to determine the level of risk that is appropriate and matched your own needs. Typically this considers how much money and resource will be sunk into work on something that was accepted but ultimately doomed to fail. 
You also define the level of risk that you reject a true effect ($\alpha$) and likewise this should be optimised to ones needs. Typically this considers the missed opportunities (what ROI would have been achieved) if it was true but rejected.
What if you haven’t powered?
Then you have no idea how well optimised the data is to answer the questions you want to ask of it. You then open up your dataset and from that point on you risk multiple comparison issues and optimising your analysis for the data rather than the problem. 
If the questions are ill formed then the data analysis may be more exploratory than confirmatory in nature but it is still useful to power so that you have confidence that you will detect a certain size of effect while balancing the impact of false positives and false negatives. 
