Does the Hadith "Were it not for Eve, no female would ever betray her husband" refer to Eve's failure to advise Adam, and does Adam's failure to advise her constitute a defect in his qawamah (guardianship/leadership)? And what is the reason for men betraying their wives?

The intent of the hadith is to explain that Eve beautified the act of eating from the tree for Adam, and it has no relation to giving advice. Treachery does not necessitate a change in lineage; heedlessness, absence of faith, and lack of fear of Allah can lead to disobedience. Satan approaches a servant through the doors of heedlessness, desire, and anger. He tempted Adam despite his forbearance and intellect, so how much more those beneath him? Satan cannot overcome a believer except when the believer is heedless.