You can. By using partitioned-regression results (F-W-L theorem), if the model includes a constant term,
$$y = \alpha + \mathbf x' \beta + u$$
then the "slope coefficients" are in practice calculated by OLS as
$$\hat{\beta}_{OLS} = (\hat {\tilde X}'\hat {\tilde X})^{-1}\hat {\tilde X}'\hat {\tilde y}= \beta + (\hat {\tilde X}'\hat {\tilde X})^{-1}\hat {\tilde X}'u$$
Where $\hat {\tilde X} = X - \bar X$ (sample mean) and for later use $ {\tilde X} = X-E(X)$. Again, here $X$ does not include a series of ones (but the model does). In other words, whether you demean a priori or not, OLS will demean the variables automatically to estimate the slope coefficients, if you also include a constant term in the model.
If you multiply and divide the above by sample size, you get the multivariate analog of $(2)$ since the variables are now in sample mean-deviation form and so these are estimated variance and covariance matrices (and in $(2)$ you should use hats, by the way).
Then
$$\text{plim}\big( \hat{\beta}_{OLS} -\beta\big) = \text{plim}\left (\frac 1n\hat {\tilde X}'\hat {\tilde X}\right)^{-1}\text{plim}\left(\frac 1n \hat {\tilde X}'u\right)$$
The Law of Large Numbers does not "jump from probability limits to expectations": it is the very essence of the Law that the probability limit is the expected value. Then (under the necessary conditions)
$$\text{plim}\big( \hat{\beta}_{OLS} -\beta\big) = E\left (\frac 1n\tilde X'\tilde X\right)^{-1}E\left(\frac 1n \tilde X'u\right)$$
Now the variables are in deviations from their true expected values, and these expected values are the true covariance matrices, and you can write for example,
$$\text{plim}\big( \hat{\beta}_{OLS} -\beta\big) = [ \text {Var}(\mathbf x)]^{-1}\cdot \text{Cov}(\mathbf x \cdot u)$$
which is the probability limit of the multivariate analogue of $(2)$.