Is statistical power relevant if there is no focus on significance/hypothesis testing? If there is no focus on NHST and P-values, then are power analyses even necessary? What if someone were to focus solely on effect sizes and confidence intervals (I know that p-values and CIs share similar statistical theories) so if someone could explain this, I'd appreciate it. 
 A: If you care about whether there's a good chance your confidence interval would include the value (or set of values) that would be under the "null" in some corresponding hypothesis test (indicating the possibility of no difference/no effect) then considering power would be important.
If you really don't care about that, then things may be different. 
You don't give enough information to tell much about what the situation is, though, so it's hard to say much more (I can't guess whether you're in a situation where you'd care; many people would, but some might not).
One way to think about using power studies to identify sample size is it gives a way to identify a sample large enough to have a good chance to say that a meaningful difference (in some sense relevant to whatever you're looking at) is not simply explainable by the action of random noise in your results.  (We see a big difference and it can't easily be dismissed as a chance effect.)
A: As @Glen_b pointed out, it depends on the nature of your problem. However that said, I often find that it's still useful to keep track of the p-value as a way to easily detect and discard seemingly-large effect sizes which aren't likely to replicate. You can do this with confidence intervals too of course, but that requires asking a more complicated question ("does the baseline value fall inside this region") vs a simple "is p < cutoff".
If you're just running a single test, then this doesn't really matter, but on those occasions where I've cared primarily about effect size as opposed to significance, it's generally because I'm running a very large number of tests, and want to reduce each one down to a single number. However even keeping confidence intervals on hand in this case, you would be misled due to the problem of multiple testing, and would be unable to correct for it without knowing the p-values. 
So in short, yes, it does still help to keep p-values handy. 
