I think doing regression via SEM is bogus. I mean, it is cute to show that you can express a linear regression as a special case of SEM, just to show how general SEMs are, but doing regression with an SEM is a waste of time, as this approach does not utilize the many advances in regression modeling specific to linear models. This is the right tool for the job issue: if nothing else is at hand, I would hammer a nail into a drywall with the screwdriver by holding the latter at a sharp end and hitting the nail with the handle, but I won't recommend doing that, in general.
In SEM, you model the covariance matrix of everything: the regressors and the dependent variable. The covariance matrix of the regressors have to be unconstrained. The covariances between the dependent variable and the regressors is what generates the coefficient estimates, and the variance of the dependent variable, the $s^2$. So you've utilized all the degrees of freedom (number of covariance matrix entries), and that's why you see a zero. You should still be able to find $R^2$ in your output, but it will be hidden deeply somewhere, not thrown at you as in regression output: from the point of view of SEMs, your dependent variable is nothing striking, you may have a few dozen regressions in your output, and you can get all of their $R^2$s, or reliabilities, somewhere, but you may have to ask for it with some TECH options in Mplus.
The missing value stuff is even greater bogus. Typically, you have to assume some sort of a distribution, such as multivariate normal, to run a full information maximum likelihood. This is very doubtful for most applications, e.g. when you have dummy explanatory variables.
The advantage of doing regression properly with R or Stata is that you will have access to all the traditional diagnostics (residuals, leverage, etc. on influence; collinearity, nonlinearity and other goodness of fit issues), as well as additional tools for better inference (sandwich estimator that can be made robust for heteroskedasticity, cluster correlation or autocorrelation). SEM can offer "robust" standard errors, too, but they don't work well when the model is structurally misspecified (and that's what one of my papers was about).
Best Answer
Significance tests of path coefficients are computed based on unstandardized estimates, and not on standardized estimates (and when they are, they should not be trusted). I don't think there is a way to impose constraints on standardized estimates in Mplus, nor should you need to. Therefore, you do need to constrain the unstandardized estimates to equality, and do a nested-model comparison where you compare this constrained model to a model in which both (unstandardized) paths are freely estimated.
The one exception where you can constrain standardized paths is when you constrain covariances between latent variables and both latent variables have a fixed variance of 1; these covariances are indeed correlations, and correlations are standardized estimates.