Reordering Static Effects with Dependencies on a Changing Board State
Yes, the existence of an object with no abilities can change the order in which effects are applied.

In the section about dependencies, rule 613.7a says


  An effect is said to “depend on” another if (a) it’s applied in the same layer (and, if applicable, sublayer) as the other effect (see rules 613.1 and 613.3); (b) applying the other would change the text or the existence of the first effect, what it applies to, or what it does to any of the things it applies to; and (c) neither effect is from a characteristic-defining ability or both effects are from characteristic-defining abilities. Otherwise, the effect is considered to be independent of the other effect.


The highlighted section clearly indicates that the dependency order of effects depends not only on the effects themselves and the objects those effects are on, but also the full set of objects that those affects could apply to. So a change to that set can change how effects are applied.

So, your description of how that example would play out is correct. The existence of the Saproling token changes the dependency order between Life and Limb and Blood Moon, which results in Gingerbread Cabin ending up with different characteristics.