If one denies the transitivity of ground, then it seems much of the motivation for the well-foundedness of ground is lost. Roughly, well-foundedness is supposed to prohibit chicken-and-egg (grounding loops) and turtles-all-the-way-down (infinite grounding chains) scenarios. But without transitivity, the former are perfectly permissible and the main intuition against the latter is weakened.
Suppose we have the following grounding loop, where the arrow represents ‘grounds’ (the letters can be alternating chickens and eggs):
A ⟶ B
↖ C ↙
This loop violates the asymmetry and irreflexivity of ground only if grounding is transitive. Sans transitivity, nothing is wrong with grounding loops of this sort.
Suppose we have the following grounding chain, where F is a fundamental being (Grandfather Turtle) and ds are derivative entities (turtle babies):
A main intuition backing well-foundedness is that concrete reality is structured roughly like the above. For example, Jonathan Schaffer says proponents of well-foundedness maintain that there must be a fundamental ground of being, for “if one thing exists only in virtue of another, then there must be something from which the reality of the derivative entities ultimately derives” (“Monism: The Priority of the Whole,” p. 37). But if the reality of dn is ultimately derived from F, we need transitivity to get us from the reality of F through the reality of d1–d3 to dn.
This is a problem for grounding theorists like Schaffer, who accept well-foundedness but reject transitivity. Of course, I don’t mean to imply that this is an irremediable problem. But it suffices to show that, at the very least, some extra machinery is required.