$\min(x)/x$ is usually not a weight. Weights are either frequency weights (indicating a count) or they are inverse-probability weights. Specifically, there are three kinds of probability weights: the probability for missing data, the probability of receipt of a treatment, or the probability of sampling in a survey design. Weights can never be negative. Software punts back an error when you try to do this. Since you believe $\min(x)/x$ is an acceptable weight, I would point out if $x<0$ and $x>0$ for any two values, this approach should appropriately fail regardless of whether the weights "make sense".
Weights are an exponential term to the likelihood, and consequently a multiplicative term to the log-likelihood and consequently to the score equations. That means that the score equations for a weighted linear regression model are:
$$S(\beta) = \mathbf{X}^T W (Y-\mathbf{X}^T\beta)$$
where $W$ is a diagonal matrix of the weights. The variance of the weighted regression model is
$$\text{var}(\hat{\beta}) = W^T \Sigma W$$ where $\Sigma$ is the covariance matrix of the residuals.
Unfortunately, applying the weights as you suggest does not result in the desired output as you describe. They obtain a difference variance basically because of how the residual variance is calculated. (I verified this by doing a little simulation to convince myself).
You then say you're confused why you get a coefficient for the weight matrix, even though you note that you input the root-weight matrix as a regressor in the model. I think you haven't realized that your software doesn't realize you mean to put this value in *in place of an intercept variable. Indeed if the model you are trying to fit is:
$$Y = \beta_0 \sqrt{w} + \beta_1 \sqrt{w}X$$
Then the syntax requires you to drop the intercept, the first coefficient to $\sqrt{w}$ would be the "intercept term where the 1s vector is mutiplied by the square-root of the weights". But again this is the wrong way to be doing things.
If your software cannot accept weights to a regression model, you need better software.