Relation between variance (covariance) and Fisher information when the estimator is a random variable

I have to answer a little question for my thesis work about statistics. I'm currently working with the following linear model:

$$u_j = x_j^{\mathrm{T}} \beta + \varepsilon_j,$$

where $u_j$ is the utility (a given data) of an alternative, $x_j$ is a vector denoting features of the alternative, $\beta$ is a parameter vector reflecting the preferences (unknown) and $\varepsilon_j$ are errors that are assumed to have independent standard Gumbel distribution.

In all the literature and works I've read, the estimation is always done by minimizing the variance (which is desirable), but in order to do that the procedure is trying to maximize the Fisher information (or a summary statistic when it is a matrix, such as the determinant or the trace), using maximum likelihood. A reference can be reviewed here.

My question is the following: the estimator $\beta$, once the data $u_j$ is fixed, is a fixed vector that can be calculated or estimated via maximum likelihood, and in that context I understand that minimizing the variance is equivalent to maximizing information. But what happens if we consider $\beta$ to be a random variable? Does it make sense to use maximum likelihood to estimate $\beta$, no matter what the coefficients are? Will there always exist that relation between covariance and information?

I'm sorry for the long text and if this question has already been asked. I searched and found nothing in this site. Lastly, if anyone can answer, could you please put some references? I need to cite this and your help will be very appreciated.