I do think constructing confidence intervals around parameter estimates (such as correlations, among other kinds) is a good thing to do. I strongly recommend it. Moreover, I don't think it should matter if the observed value is 0 or any other focal value. If someone claims that they did a study and found that the correlation was 0, you would want to know something about how confident we can be regarding that answer, and confidence intervals help to provide that information. There is a big difference between $r=0\pm.5$ and $r=0\pm.05$.
The concept of statistical power is defined within the Neyman-Pearson framework. (The Neyman-Pearson framework is often easiest to understand when contrasted with the Fisherian approach; you can find a nice, quick overview of NP from that perspective here.) If you can specify a type I error rate ($\alpha$), a sample size ($N$), and a candidate effect size ($r$), you can calculate the probability of making a type II error ($\beta$) or the probability of correctly rejecting the null hypothesis ($1-\beta$). But if you're not interested in significance testing, I recognize that this conception of power does become less appealing.
However, I gather your criticism is that $r$ will never truly equal 0 within your domain. That is pretty common with observational research as Meehl (1990) famously pointed out. Thus, testing to see if $r=0$ is testing if the underlying network of causal forces is perfectly balanced, which is typically very unlikely (but see here and here for some counterexamples). Nonetheless, you can take any point value as your null (although this almost never happens in practice). For example, you could do a one-tailed test to see if $r>.3$ (or $< -.3$), which Meehl gestimated is the level of "ambient correlational noise".
I say these things for the sake of completeness; I'm not trying to push you towards significance testing. You state that "[i]t's the relative amount that matters", a sentiment with which I agree wholeheartedly (see my answer here, for instance). There is another concept, related to power, that would better suit your needs. The framework you are looking for is known as Accuracy in Parameter Estimation, or AIPE (Maxwell et al., 2008), which I recommend. You will want to take a look at the work by Ken Kelly, he describes AIPE thusly:
Accuracy in parameter estimation in this sense is operationalized by
obtaining confidence intervals that are sufficiently narrow. A narrow
confidence interval provides more information about the population
parameter of interest than does a wide interval or a null hypothesis
significance test, as the interval reveals whether or not some null
value (generally zero) can be rejected and it defines the range of
plausible values for the parameter at some specified level of