Yes, the weights of the first eigenvector of a covariance matrix represent the market factor and also the largest source of systematic risk (variation of returns).
Why PCA? Well, PCA simply identifies the eigenvector that maximally explains the variance of the system. It turns out that this is the "market factor" - i.e. the tendency of securities to rise and fall together as an asset class. Why is this the market factor? If you examine the weights (factor loadings) of the first eigenvector in a histogram you will find they are generally all of the same sign whereas this is not the case for any of the subsequent eigenvectors (which represent sectors or style factors, that is to say other sources of systematic risk). In other words, it is empirically the case that there is a dominant systematic factor called the equity risk premium explaining the variance of returns.
Why is it empirically the case? Because macro variables such as monetary, fiscal policy, growth expectations, political risk, regulatory risk and other factors influence the returns of all stocks (although to varying degrees as measured by the security beta).
When you say "My concern is that PCA is maximizing the variance", it is more accurate to say that PCA is identifying that linear combination of security weights (i.e. an eigenvector) that has the highest variance which we call the market factor because the weights are generally the same sign. A portfolio constructed using these weights - an eigenportfolio - certainly would have high variance -- in fact it's the set of portfolio weights that produces the most volatility vs. any other set of portfolio weights since it loaded on the primary source of systematic risk. (As a side note, you could form a eigenportfolio that has minimum variance by identifying a principal component with a low eigenvalue.)
Indeed, if you plot the return of a portfolio based on the weights of the "first eigenportfolio" you will see the returns are highly correlated (but not equivalent to) the index itself as in the chart below from Marco Avellenada's paper:
There is a distinction between the market portfolio and the market factor described above. The "market portfolio" in the CAPM sense is the sum of all assets in the universe which by definition is market-cap weighted (since all assets are held in aggregate). Beta attempts to estimate the covariance of a security with respect to the market portfolio where the S&P or MSCI is a typical proxy. The market portfolio is fully diversified of idiosyncratic risk so its variance is explained by exposures to systematic risk only. The conceptual connection is that the Beta is the measure of the systematic risk or common source of variation in returns. Therefore, the first eigenvector (which identifies that linear combination of assets of maximal variance) is sometimes used as a proxy for the market portfolio. Also, a given security's "loadings" on the first principal component is one way of measuring the security's beta to the market factor.
On the second question, another way to choose the weights of the basket is market-cap weighted in line with the the theoretical definition of market portfolio. However, it's not clear what your goal is to "select weights of the basket". You correctly point out that you would simply be identifying a portfolio with pure exposure to systematic risk where idiosyncratic risk is diversified away.
