1) Regulations require that you be able to do so separately
2) You need to mix and match to do hypotheticals
3) You need to do it from the ground up to ensure accuracy since position data is ALWAYS flawed.
And one more thing: correlations are not explicitly used by most banks. Correlation is used in the variance/covariance and Monte Carlo VaR. About 86% of a survey of 70 large banks use hist sim, which only captures correlation implicitly. This is from a survey that one of my students performed for his masters degree.
For ease, consider the case of a bond portfolio (relates to point 3 from Ken), apart from the rebalancing & maturity of individual bonds, we may have bonds getting called (30 day notification period) or convertibles converted (a different risk profile). Hence bottom up aggregation is a viable measure in such portfolios.
Your concern about correlation matrix is legit and this is one of the reasons why people choose factor models and estimate systematic risk using factor covariance matrix (reduced dimensions as usually K factors < N instruments). Of course there is a full security correlation matrix required for estimating idiosyncratic risk for the complete risk profile, but then it is mostly a diagonal matrix (if there are no bonds from the same issuer).
I am confused about two points about VaR still which I hope anyone can shed some light - I think it was mentioned for Monte Carlo and parametric var the correlation matrices are used - I get the point for Monte Carlo; but for parametric VaR, is there any real difference between computing the portfolio variance off the portfolio value changes (where we define value as the summation of number of stocks * price) vs handling it through w’Cw where we use the covariance matrix?