A result of Bhargava-Skinner-Zhang says that a majority of elliptic curves over $\mathbb{Q}$ satisfy the BSD rank conjecture. 


The are infinitely many isomorphism classes of $\mathbb{Q}$ so one naturally has to be more precise about what "a majority" means; in their case, the elliptic curves are done by height.


The question is what are the arguments for the ordering by height being the correct ordering to consider (in the sense that Bhargava-style results actually constitute non-zero progress to the BSD rank conjecture)? 


The appearance of percentages in these theorems for some reason reminds me of traders on a bazaar advertising their products ("Ours can deal with 50% of the elliptic curves", "Forget him, mine can do 60%"). I just want to understand whether there is in fact a deep connection between the results in question and BSD or whether it is extraordinarily good sales pitch of the authors (this question is not meant to cast a shadow of doubt on anything other than the OP's knowledge of the subject).



Why do Bhargava-Skinner-Zhang consider the ordering by height?
Roughly speaking, the height of an arithmetic object (number, variety, ...) is a natural measure of its complexity, say in the sense of "how many bits does it take to describe the object." (This is not meant to be rigorous, but you seem to want to know why people use "heights".) One can then ask for heights (complexity measures) that have nice properties, for example, transform nicely (functorially) for maps, i.e., ht(f(object)) is related to some nice function of ht(object). Weil heights and morphisms constitute a nice example of this, and canonical heights on abelian varieties behave even more nicely. For [BSZ], counting elliptic curves by height is more-or-less counting by (1) # of bits to describe the $j$-invariant (i.e., the $\bar{\mathbb Q}$ isomorphism class of $E$) plus (2) # of bits to describe how twisted the curve is.


