Ethical Culture's peculiar position is that it is legally a religion in the United States and has no theology. That has been tested, and it has held.
In Torcaso v. Watkins in 1961 the Supreme Court struck down a Maryland requirement that officeholders declare a belief in God. In a footnote, Justice Black listed religions in the United States that do not teach belief in the existence of a god, and named Ethical Culture among them. That footnote has since been quoted very widely by opponents of secular education, as though the Court had found secular humanism to be an establishment of religion. It found nothing of the kind, the footnote is an aside illustrating that a religion need not be theistic, and the misuse is a live rhetorical dispute in American church-state argument rather than a legal holding.
Courts have upheld the tax status of Ethical Societies as religious organisations, and their Leaders solemnise marriages. The dual identity attracts objection from both directions: from theists who deny that anything without a god can be a religion, and from atheists who object to the religious framing at all.
The practical record is the point Adler himself would have made. The movement founded the Ethical Culture Fieldston School, ran district nursing and a free kindergarten, was involved in the founding of the American Civil Liberties Union, and campaigned on tenement housing and child labour. Whether that is religion is a question about definitions. That it was done is not.