Degrees of freedom in multiple regression thought of as a path analysis using standardized variables

I keep reading that multiple regression is "just-identified" (df = 0) when viewed as a path analysis. If I'm using unstandardized variables, I believe this. For example, with five variables—4 predictors (IVs) and 1 response (DV)—there are 5(5 + 1)/2 = 15 variances and covariances available to me as data. Then the model needs to estimate 4 variances for exogenous variables, 6 covariances between exogenous variables, 4 paths from IVs to DV, and 1 error variance for a total of 4 + 6 + 4 + 1 = 15 parameters. Okay so far.

Now suppose I want to use standardized variables. Because all variances are now 1, I only have 10 correlations to use as data. On the parameter side of things, I no longer have to estimate 4 variances for the exogenous variables (those are set to 1), but I still have the 6 correlations between exogenous variables, the 4 paths from IVs to DV and the 1 error variance. But now I count 6 + 4 + 1 = 11 parameters. Ruh, roh.

What am I missing here?

If you know the four regression paths and you know the correlations between predictors, you know $R^2$. If you know $R^2$, and you know the variance of the outcome (it's 1.00), you know the error variance and so this does not need to be estimated.