This definition of connectedness is more immediately suited for proving disconnectedness rather than connectedness. Basically, this definition really defines what it means to be disconnected, and then says that something is connected if it is not disconnected. This is very different than path connectedness, where the claim is that for any two points there is a connecting entity, namely a path, between the two points. Such a definition is more immediately suited to proving connectedness. If topological connectedness could be defined, equivalently, in a manner more like path connectedness, by exhibiting a connecting entity, then it would be useful for proving connectedness.
For metric spaces there is such a reformulation of connectedness. A scale on a metric space $X$ is a choice of a positive real number $R_x$ for each point $x\in X$. Given a scale, a walk is a sequence of finitely many points such that for any two adjacent points in the sequence $x,y$, either $d(x,y)\le R_x$ or $d(y,x)\le R_y$. It then holds that $X$ is connected in the topological sense if, and only if, for any two points and any scale on $X$ there exists a walk between these two points.
To obtain a similar characterisation for arbitrary topological spaces use the fact that every topological space is metrisable, provided the metric is allowed to take values in structures more general than the non-negative real numbers (and dropping the requirement that the metric is symmetric). This formalism was developed by Flagg, allowing the metric to take values in a value quantale (Quantales and continuity spaces). This approach was shown to be categorically equivalent to standard topology (A note on the metrizability of spaces). The adaptation of the notion of connectedness from classical metric spaces to Flagg's formalism was done in Metric characterisation of connectedness for topological spaces, where classical connectedness and uniform connectedness are shown to be two instances in a whole hierarchy of notions of connectedness (to which path connectedness does not belong). Among other things, some classical results about connectedness are proved using the new formalism.