Should we always minimize squared deviations if we want to find the dependency of mean on features? It is known that for any distribution a minimization of squared deviations from a value $c$ gives mean of the distribution. In other words, if we generate many values $y_i$ using our distribution, and then try to find $c$ that minimizes the following expression:
$
\lim_{N \to \infty}  \frac{1}{N}\sum_{i}^N (y_i - c)^2
$
then we find that the minimum is achieved when $c = \mu$, where $\mu$ is the mean of the given distribution. And for any $N$ the above minimization gives the sample mean.

Now, let's assume tat we have a distribution which depends on a single real-valued feature $x$, and we have a data-set generated by this distribution: ($x_1$, $y_1$), ($x_2$, $y_2$), $\dots$, ($x_n$, $y_n$). Then we try to find a function that gives dependency of the mean of this distribution on the feature:
$
\mu = \mu(x, p_1, p_2, \dots, p_k)
$
where $p_i$ are model parameters that we want to find.
Obviously, that we can get the desired function (values of the parameters) if we choose them by minimization of squared deviations:
$
\frac{1}{N} \sum_{i}^{N} [y_i - \mu(x_i, p_1, p_2, \dots, p_k)]^2
$
In other words, we can find the correct answer (correct values of the parameters $p_i$) under condition that $N$ goes to infinity and that the real function, that we are searching, is covered by the space of functions that we consider.
To summarize, the minimization of squared deviations can give not only mean of a given constant distribution, it can also gives correct dependency of the mean on a feature.

In reality $N$ is never infinity and can be rather small. As a result minimization of squared deviation will not give exactly correct values of the model parameters $p_i$. For another sample of the same size we will give other estimations of the parameters $p_i$. In general case, we can say that our estimations of the parameters are somehow distributed.
If we, instead of squared deviations minimize some other measure, we will get other distributions of our model parameters $p_i$. So, different measure of accuracy give different distributions in the parameters space for a given fixed sample size.
These distributions are characterized by their "width" and possible a "shift" relative to the correct point (true values of the model parameters).
Can it be the case that a use of something different from squared deviation gives a better estimate of model parameters (for example more accurate (smaller width) and with smaller or no systematic from the correct model parameters?

Minimizing the MSE in the cases you describe indeed produces a consistent estimator for the model parameters. The consistency is related to the fact that the derivative of the MSE, and therefore the first order optimality condition, is linear in the observations $y_i$. So, if you want a consistent estimator without any additional information on the distribution of the samples, MSE is the only option.
On the other hand, if you have a parametric form for the distribution of $y_i$, then the maximum likelihood estimator (MLE) is also consistent, but also asymptotically unbiased and efficient, meaning that it has the smallest variance among all unbiased estimators (in the limit $n \to \infty$). In that sense it is the most "accurate" estimator possible.
