Can COC force a soldier to go to the promotion board and separate if they fail?

The reg is AR 600-8-19.
"Roughly 3 years" is primary zone, assuming this Specialist pinned on 12 months ago, there is only a TIG requirement, no TIS requirement (table 7-4)
This Specialist can get a Bar to Reenlistment now if they do not have SSD1/DLC1 complete. That bar would be lifted upon completion.
If the Specialist appears and fails a board, or fails to board,:
3-1 d.:
First-line leaders must formally counsel (in writing) Soldiers upon attainment of primary zone eligibility who are not integrated into the promotion recommend list. Counseling must identify what Soldiers must do to improve their knowledge, skills, and attributes and to prepare themselves for increased responsibility (refer to para 1 – 28). The counseling also must address the consequences of not being integrated into the promotion recommended list. Those consequences include a potential bar to continued service (AR 601 – 280) because the Soldier is noncompetitive for promotion to the next
higher grade and does not demonstrate leadership potential. Additionally, retention control points (AR 601 – 280) used to determine the maximum years of service a Soldier may serve without promotion should be incorporated into the counseling.

In short, they cannot separate an E4 for failing a promotion board, but a bar to reenlistment is a possible course of action after failing two or declining a second one.

They can make a SM go to the board but there would have to be something heavier to try to force them out. A board failure wouldn't be enough to warrant that.