Kalman Filter alternative derivation Take a state space model of the form
$y_t=\alpha_t+\epsilon_t$ where $\epsilon_t\sim NID(0,\sigma_{\epsilon}^2)$
$\alpha_{t+1}=\alpha_t+\eta_t$ where $\eta\sim NID(0,\sigma_{\eta}^2)$
Futhermore, it is assumed that $E(\epsilon_t\eta_s) = 0$, for all $t,s$, that is, the error terms are independent. More, error tersma re also independent of $\alpha_1$, the initial state.
I do not get why we would apply this restrictive assumption for the Kalman Filter when we could easily estimate other parameters via MLE. For instance, suppose that $\mathbb{C}ov(\epsilon\eta) = I\sigma_{\eta,\epsilon}$ so that $E(\epsilon_t\eta_s) = 0$ for all $t\neq s$ and $E(\epsilon_t\eta_s) = \sigma_{\eta,\epsilon} \neq 0$.
We are interested in estimating $a_{t+1}=\mathbb{E}(\alpha_{t+1}|Y_t)$ and $P_{t+1}=\mathbb{V}ar(\alpha_{t+1}|Y_t)$, where $Y_{t}=(y_1,y_2,...,y_t)'$. Furthermore, I follow Durbin and Koopman textbook in that $a_{t|t}=\mathbb{E}(\alpha_{t}|Y_t)$ and $P_{t|t}=\mathbb{V}ar(\alpha_{t}|Y_t)$.
Now, $a_{t+1}=\mathbb{E}(\alpha_{t+1}|Y_t) =\mathbb{E}(\alpha_{t}+\eta_t|Y_t)=\mathbb{E}(\alpha_{t}|Y_t)=a_{t|t}$ and $P_{t+1}=\mathbb{V}ar(\alpha_{t+1}|Y_t)=\mathbb{V}ar(\alpha_{t}+\eta_t|Y_t)=\mathbb{V}ar(\alpha_{t}|Y_t) + \mathbb{V}ar(\eta_{t}|Y_t) + 2 \mathbb{C}ov(\eta_t,\alpha_t|Y_t)=P_{t|t}+\sigma_{\eta}^2+2 \mathbb{C}ov(\eta_t,\alpha_t|Y_t)$. 
Under independence, $\mathbb{C}ov(\eta_t,\alpha_t)=0$. However, under my assumption, $\mathbb{C}ov(\eta_t,\alpha_{t-1}+\eta_{t-1})=\mathbb{C}ov(\eta_t,y_t-\epsilon_t)=\mathbb{C}ov(\eta_t,-\epsilon_t)=-\sigma_{\eta,\epsilon}$, leading to:
$P_{t+1}= P_{t|t}+\sigma_{\eta}^2-2\sigma_{\eta,\epsilon}$.
Now, since all distributions are normal, $P_{t+1}$ and $a_{t+1}$ are easily found if we find the expression for the pdf of $\alpha_{t}|Y_t$ - name it $f(\alpha_{t}|Y_t)$. Then,$f(\alpha_{t}|Y_t)=f(\alpha_{t}|Y_{t-1},y_t)=\frac{f(\alpha_{t},y_t|Y_{t-1})}{f(y_t|Y_{t-1})}=\frac{f(\alpha_{t}|Y_{t-1})f(y_t|Y_{t-1},\alpha_{t})}{f(y_t|Y_{t-1})}$.
Since these pdf's are fully defined by their 1st and 2nd moments, we just need to compute those as:
$\mathbb{E}(y_t|Y_{t-1})=\mathbb{E}(\alpha_t+\epsilon_t|Y_t)=a_t+0=a_t$
$\mathbb{V}ar(y_t|Y_{t-1})=\mathbb{V}ar(\alpha_t+\epsilon_t|Y_t)=\sigma_{\epsilon}+P_{t|t}$
$\mathbb{E}(y_t|Y_{t-1},\alpha_t) = \alpha_t$
$\mathbb{V}ar(y_t|Y_{t-1},\alpha_t) = \sigma_{\epsilon}$.
Substituting in the pdf's above and computing yields:
$f(\alpha_t|Y_t)=N(a_t+\frac{P_t}{P_t+\sigma_{\epsilon}^2}v_t,\frac{P_t\sigma_{\epsilon}^2}{P_t+\sigma_{\epsilon}^2})=N(a_{t|t},P_{t|t})$, where I define $v_t=y_t-a_t$, from which it follows that:
$a_{t+1}=a_{t|t}=a_t+\frac{P_t}{P_t+\sigma_{\epsilon}^2}v_t$
and
$P_{t+1}=P_{t|t}+\sigma_{\eta}^2-2\sigma_{\eta\epsilon}=\frac{P_t\sigma_{\epsilon}^2}{P_t+\sigma_{\epsilon}^2}+\sigma_{\eta}^2-2\sigma_{\eta\epsilon}$
Then, the Kalman Filter becomes,
$v_t = y_t-a_t$
$F_t = \mathbb{V}ar(v_t|Y_{t-1})=P_t+\sigma_{\epsilon}^2$
$a_{t|t}=a_t+K_t v_t$
$P_{t|t}=P_t(1-K_t)$
$a_{t+1} = a_{t|t}$
$P_{t+1} = \frac{P_t\sigma_{\epsilon}^2}{P_t+\sigma_{\epsilon}^2}+\sigma_{\eta}^2-2\sigma_{\eta\epsilon}$ for $t=1,...,n$ and $K_t = P_t/F_t$ denotes the Kalman Gain. Then $\sigma_{\eta\epsilon}$ could be estimated via MLE together with the other hyperparameters $\sigma_{\eta}$ and $\sigma_{\epsilon}$.
Does this make sense or am I missing something here? Assuming the derivation is correct, would there be any disadvantage of estimating $\sigma_{\eta\epsilon}$ via MLE?
Thanks 
 A: You are right in that the covariance parameter could be estimated by maximum likelihood. However, care is needed because including this additional parameter may lead to a problem of identification of the parameters.
For example, let's take the local level model defined as follows:
\begin{align}
&y_t = m_t + \epsilon_t \,, &\epsilon_t \sim NID(0, \sigma^2_\epsilon) \\
&m_t = \mu + m_{t-1} + \eta_t \,, &\eta_t \sim NID(0, \sigma^2_\eta) \\
&E(\epsilon_t \eta_s) = \sigma_{\epsilon\eta} &\hbox{ if } t = s \hbox{ and } 0 \hbox{ otherwise} \,.
\end{align}
It can be checked that the autocovariances of this model are:
\begin{align}
\gamma(0) &=  2\sigma^2_\epsilon + \sigma^2_\eta + 2\sigma_{\epsilon\eta} \\ 
\gamma(1) &= -\sigma^2_\epsilon - \sigma_{\epsilon\eta} \\
\gamma(k) &= 0 \,, \quad \hbox{for } k > 1 \,.
\end{align}
Given the sample autocovariances $\gamma(k)$, there is no unique solution to the system of equations, since we have two equations to be solved and three parameters ($\sigma^2_\epsilon$, $\sigma^2_\eta$ and $\sigma_{\epsilon\eta}$).
The common restriction $\sigma_{\epsilon\eta} = 0$ can therefore be interpreted as an identifying restriction.

Morley et al. (2003) [1] is an interesting discussion on this issue in the field of economics. The authors fit a model consisting of a trend and a stationary cycle with and without the zero-correlation restriction between the components.
[1] James C. Morley, Charles R. Nelson and Eric Zivot (2003).
"Why Are the Beveridge-Nelson and Unobserved-Components Decompositions of GDP so Different?". The Review of Economics and Statistics. Vol. 85, No. 2.
URL http://research.economics.unsw.edu.au/jmorley/mnz03.pdf.
