There may be situations I am unaware of, but I don't think the standard setting is to have a metric space $(X,d)$, where you know $X$ is connected under $d$, and that there are at least two distinct points in $X$, but don't already know the space is uncountable (and care!). I think it would be far more likely to know that $X$ is at most countable, and then we would know the space must be disconnected, regardless of the metric we choose to use.
For example, for those that haven't seen Ostrowski's theorem, and have no idea what metrics can be placed on $\mathbb{Q}$, your result immediately shows it is impossible to construct a metric under which $\mathbb{Q}$ is a connected metric space. (That's not to say it's a bad idea to get your hands dirty, try to build a metric d' so that (\mathbb{Q},d') is a connected metric space, and see what goes wrong!)
One could then see this as an argument to construct $\mathbb{R}$ from $\mathbb{Q}$, since no matter what metric we use, there are holes.
I suppose one could also say, if there is a topology $\tau$ on $\mathbb{Q}$ so that $(\mathbb{Q},\tau)$ is a connected topological space, then we also know that this space is not metrizable. I don't know if this is a particularly useful point of view though..
Of course these are only examples using $\mathbb{Q}$ to illustrate the point, and the same holds for far more odd 'looking' at most countable spaces, where it may be far less intuitive that there are no metrics to make the space a connected metric space.