I agree with what you wrote, but I think that critics of ID try to have their cake and eat it too and to me it looks as self-defeating as ID.
If ID is a religious notion and has a “particular religious claim” that God intervened in nature such that complexity points to an intelligent designer, how can the scientific evidence conflict with or refute that? Evolutionary science cannot say that complexity could not have been designed/created de novo in some way in the process rather than evolved because we were not there.
I do not understand how ID critics aren’t sowing harmful conflict if they don’t acknowledge this, if they believe it is a religious argument.
IMO, the criticism of the current theory of evolution that ID proponents make should not be confused with arguments FOR intelligent design, or ID does become a scientific argument and not a religious one and ID critics have just handed ID their argument.
But I think it’s too late for that - it has already been and is being responded to as science. If so, drop the “religious claim.”