I think there are simple practical reasons. Rectifiable sets are often used to integrate wrt. the corresponding Hausdorff-measure, so having an extra null-set does not matter. Also a common way to prove rectifiability is by exhaustion, i.e. cover 1% (in measure) of the set by a Lipschitz map and then iterate. This procedure naturally leaves over a null-set.

As a comment on that comment, especially in the context of Federer, $BV$ can be considered a special case of normal currents. Ambrosio and Kirchheim then later extended those to metric spaces, though not in the precise way asked for in the question. The general problem is that it is not a-priori clear what the volume of a jump is if there is no simple linear interpolation between points. The idea of cartesian currents (Giaquinta, Modica & Soucek) should provide a way, but afaik has never been generalized to metric spaces. So I am not sure if there is a way to answer without writing 2-3 papers.

I would even say that hearing is not a good sense to encode mathematics at all. A long formula can usually be read at a relatively short glance even if it involves double fractions and multiple indices, but dictating something like this in an unambiguous way would be a serious exercise in patience.

To add to this, a similar definition is quite common in the theory of minimal surfaces, which often deals with non-compact surfaces (e.g. Simon's cone). Also the name "local minimizer" is normally used to talk about functions $f$ that minimize the full integral in some neighborhood of $f$ in $X$.

When looking at that list, there is an abundance of events that are more computer science than mathematics. Considering that the 1960's was about the decade when computers became widespread enough that most universities had access to some, this might be an explanation for the bump. Also in general, as defining "breakthrough" is hard, I believe that people who create such a list, will tend to even things out to make such a list more uniform, so I don't think it is that significant of a source.

I don't have any special reference in mind, but the names I would associate with these ideas are Fusco, Maggi and Pratelli and their respective coauthors, so that might be a start. Though most of their results probably would need to be adapted a bit to fit in your context.

At least for a sufficiently nice looking $\Omega$, there might be some information in the excess volume as well. In the worst case, $\partial \Omega \cap B_\delta$ might be the area-minimizer for a fixed volume $\Omega \cap B_\delta$ (which also involves curvature as a Lagrange-multiplier), while your $[\Omega_\delta]$ is a minimizer without such a volume constraint. There is a sheer endless amount of literature on (reverse, relative) isoperimetric inequalities that could provide some ideas here, in particular since they often quantify the error in terms of additional quantities.

I guess a quick $O(n)$ heuristic would be to pick the hyperplane that is normal to the vector pointing from the origin to the center of mass. I can think of some pathologic configurations where this fails spectacularly, but for more generic configurations, it should yield a good first approximation.

You will definitely need some sufficient notion of niceness, with the obvious standard counterexample being the open, dense sets of arbitrary small measure you can construct as the union of a sequence of smaller and smaller balls around any countable dense set. (Btw, this set should really have a name, since it comes up so often in similar questions...)

@Riku I have edited what I think the solution should be but there may be some mistakes in there.