Suppose you have a Normal Mixture Model with 2 Components - you could write this model as follows:
$\pi_1 N(\mu_1, \sigma_1) + \pi_2 N(\mu_2, \sigma_2)$
In the above model, there are 6 unknowns : $\pi_1$, $\mu_1$, $\sigma_1$, $\pi_2$, $\mu_2$, $\sigma_2$. In the context of Mixture Models, these 6 unknowns are generally estimated iteratively using the EM (Expectation Maximization) Algorithm.
My Question: What is preventing us from writing the above model in "Likelihood Format" as a function : $L(\pi_1, \mu_1, \sigma_1, \pi_2, \mu_2, \sigma_2) = \dots$ and then jointly estimating these 6 unknowns by either taking the partial derivatives of the likelihood function relative to each of these 6 terms and solving these equations - or using some optimization algorithm like Gradient Descent (if there is no closed form solution)?
Why has the EM algorithm become the standard algorithm for estimating the unknowns of Mixture Models?