Why the arrow needs an explanation
Temporal order and the thermodynamic arrow answer different questions. A complete history can contain a direction without requiring a moving present.
A temporal dimension does not, by itself, explain the asymmetric processes through which we experience time. A description can place one event earlier than another while leaving unanswered why familiar macroscopic processes have a characteristic direction.
The thermodynamic arrow identifies a direction associated with increasing entropy. In their discussion of its cosmological origin, Carroll and Chen connect the observed increase to the low entropy of the early universe. They propose a cosmological mechanism involving inflation; the proposal is a proposed explanation, not an established origin story.
For an eternalist, this distinction is important. A complete history can contain an entropy gradient. Its completeness does not make every temporal region physically alike. Equal existence and equal physical condition are independent claims. Nothing prevents different parts of an existing history from having profoundly different properties.
Nor does the gradient prove that a universal present advances along it. An asymmetry within a history and an ontological process that adds events to reality would be different features. A theory of becoming can appeal to the arrow as a motivation, but must explain why the asymmetry requires becoming rather than being describable through temporal relationships.
The low-entropy condition also leaves a further question. Identifying a boundary condition that helps explain familiar irreversibility does not explain why that condition obtains. Cosmological proposals attempt to address that additional burden. The explanation can reach beyond everyday processes without thereby being complete.
This gives the architecture of time two distinct tasks. One concerns which events exist and how they are temporally related. The other concerns why the history has the asymmetries we inhabit. Eternalism is a serious candidate for the first task. It cannot perform the second merely by announcing that the block is complete.
The distinction makes the picture richer. A reality can be temporally extensive and internally directional. Its inhabitants may be situated within conditions that constrain memory, prediction, and action. Explaining those constraints will tell us more about our place in the history than a general declaration that time is an illusion.