This is a good question, but as it appears from it that you know PCA and CCA a deal, so you are able to answer it yourself. And you do:
[CCA] builds the canonical variates not to blindly [wrt the existence
of X] maximize explained variance [in Y], but already with the final
purpose of maximizing correlation with X in mind.
Absolutely true. The correlation of the 1st Y's PC with X set will almost always be weaker than the correlation of the 1st Y's CV with it. This comes apparent from pictures comparing PCA with CCA actions.
The PCA+regression you conceive of is two-step, initially "unsupervised" ("blind", as you said) strategy, while CCA is one-step, "supervised" strategy. Both are valid - each in own investigatory settings!
1st principal component (PC1) obtained in PCA of set Y is a linear combination of Y variables. 1st canonical variate (CV1) extracted from set Y in CCA of sets Y and X is a linear combination of Y variables, too. But they are different. (Explore the linked pics, also pay attention to the phrase that CCA is closer to - actually a form of - regression than to PCA.)
PC1 represents set Y. It is the linear summary and the "deputy" from set Y, to face the outer-world relationships later (such as in a subsequent regression of PC1 by variables X).
CV1 represents set X within set Y. It is the linear image of X belonging to Y, the "insider" in Y. The Y-X relationship is already there: CCA is a multivariate regression.
Suppose I've got a children sample's results on a school anxiety questionnaire (such as Phillips test) - Y items, and their results on a social adaptation questionnaire - X items. I want to establish the relationship between the two sets. Items of both inside X and inside Y correlate, but they are quite different and I'm not pleased with the idea to bluntly sum up item scores into a single score in either set, so I'm choosing to stay multivariate.
If I do PCA of Y, extracting PC1, and then regress in on X items, what does it mean? It means that I respect the anxiety questionnaire (Y items) as the sovereign (closed) domain of phenomena, which can express oneself. Express by issuing its best weighted sum of items (accounting for the maximal variance) which represents the whole set Y - its general factor/pivot/trend, "mainstream school anxiety complex", the PC1. It is not before that representation is formed that I turn to the next question how it might be related to social adaptation, the question I'll check in the regression.
If I do CCA of Y vs X, extracting the 1st pair of canonical variates - one from each set - having maximal correlation, what does it mean? It means that I suspect the common factor between (behind) both anxiety and adaptation which makes them correlate with each other. However, I have no reason or ground to extract or model that factor by means of PCA or Factor analysis of the combined set "X variables + Y variables" (because, for example, I see anxiety and adaptation as two quite different domains conceptually, or because the two questionnaires have very different scales (units) or differently-shaped distributions that I fear to "merge", or the number of items is very different in them). I'll be content with just the canonical correlation between the sets. Or I might not be supposing any "common factor" behind the sets, and simply think "X effects Y". Since Y is multivariate the effect is multidimensional, and I'm asking for the 1st-order, strongest effect. It is given by the 1st canonical correlation and the prediction variable corresponding to it is the CV1 of set Y. CV1 is fished out of Y, Y is not selbständig producer of it.
Best Answer
The first step of ICA is to use PCA and project the dataset into a low-dimensional latent space. The second step is to perform a change of coordinates within the latent space, which is chosen to optimize a measure of non-gaussianity. This tends to lead to coefficients and loadings that are, if not sparse, then at least concentrated within small numbers of observations and features, and that way it facilitates interpretation.
Likewise, in this paper on CCA+ICA (Sui et al., "A CCA+ICA based model for multi-task brain imaging data fusion and its application to schizophrenia"), the first (see footnote) step is to perform CCA, which yields a projection of each dataset into a low-dimensional space. If the input datasets are $X_1$ and $X_2$, each with $N$ rows=observations, then CCA yields $Z_1 = X_1W_1$ and $Z_2 = X_2W_2$ where the $Y$'s also have $N$ rows=observations. Note that the $Y$'s have a small number of columns, paired between $Y_1$ and $Y_2$, as opposed to the $X$'s, which may not even have the same number of columns. The authors then apply the same coordinate-changing strategy as is used in ICA, but they apply it to the concatenated matrix $[Z_1 | Z_2]$.
Footnote: the authors also use preprocessing steps involving PCA, which I ignore here. They are part of the paper's domain-specific analysis choices, rather than being essential to the CCA+ICA method.