Is there empirical evidence on usability of Cobb-Douglas production function?
The simple answer to why the Cobb-Douglas functional form is used is because it is at least a log-linear approximation to some higher-order production function. That is, suppose you take a functional form that looks like this: $\log Y_t = f(A, K, L)$. Then a linear approximation would look like the Cobb-Douglas production function. (For a small $1\%$ increase in $K$, we get approximately an $\alpha \%$ increase in $Y$.)

One potential source of the strange estimation results you're getting is likely a bias due to the endogeneity of the regressors. Total factor productivity (TFP) in your notation is $A$. It is changing over time. In your regression, it shows up in the error term. Changes in productivity affect the composition of the inputs $L$ and $K$. Thus, $L$ and $K$ are correlated with error term. To see this, solve any typical cost minimization for a firm. $L$ and $K$ will depend on $A$. Also note that the behavior will be different in the short-run vs the long-run.

Hope this helps!

EDIT:

If you're interested, here is a problem set (no solutions available) that walks you through a demonstration of the bias described above. In the end, one potential improvement to the estimation procedure is explored via Monte Carlo simulation. This improvement relies on the availability of panel data of firm input/output data.