Ever play scissor paper rock...?
Sometimes if A is greater than B and B is greater than C then certainly A is greater than C. This is true if we compare something like mathematical amounts (like weight).
However, it sometimes works out that A is greater than B; B is greater than C; and C is in fact greater than A!
Scissors cuts paper, and paper covers rock. Now, shouldn't scissors beat rock, because it beats the thing that beats rock? NO! Rock blunts scissors.
The Gemara for instance asks that strangulation be considered more severe than burning. Burning was singled out for a daughter of a cohen; But strangulation was singled out for someone who strikes a parent. The Gemara was able to show that since a daughter of a cohen got her punishment upgraded from strangulation to burning, then obviously burning is the more severe punishment.
If we didn't go through that logical step, we may have come to the conclusion that burning is more severe than the sword and the sword is more severe than strangulation, but perhaps strangulation is more severe than the sword?? The same logic applies to stoning if we didn't walk through the steps of proving it was more severe than each of the other three.
The difference of course would be concerning someone who was convicted of two sins. One deserving of burning and the other deserving of strangulation. You can only give the condemned the death which is more severe. Which one?