The essence of the quote in question is “failure ... might ... betray ... claim to be feeling fear or distress.” Evidently some person has claimed that they felt fear or distress. The truth of that claim might be brought into question, or have doubt cast upon it, or be betrayed, or be revealed as false, if some set of actions were not done.
A “claim to be feeling fear or distress” is like a declaration that one feels afraid. For example, if person E says “I am afraid, fearful, and thoroughly distressed”, then E claims to be feeling fear or distress.
Among other senses of the verb betray, consider the following.
• To prove faithless or treacherous to, as to a trust or one who trusts; to be false to; to deceive; as, to betray a person or a cause
• To disclose or discover, as something which prudence would conceal; to reveal unintentionally
• To show or to indicate; -- said of what is not obvious at first, or would otherwise be concealed
The first of these senses is what is used in the quoted passage. Since the absence of actions does not disclose, discover, reveal, show, or indicate that distress was felt, the latter two senses do not apply. Instead, the absence gives the lie to or is faithless to the claim to be feeling distress, and thus betrays the claim.