When President Donald Trumpintroduced his Supreme Court pick on live television last week, he said Neil Gorsuch had “demonstrated a commitment to helping the less fortunate” by working in the Harvard Prison Legal Assistance Project and the Harvard Defenders.

His affiliation with these volunteer programs—which offer law school students real-life legal experience representing prison inmates and the poor—helped give Mr. Gorsuch’s deeply conservative résumé a personal touch, and the groups were highlighted in news reports about...