As its known in verse 4:34, the husband has the responsibility over his wives.
In a case where A husband is not able to convince his wife to stop from a sin in which he is aware about, does this imply a sin towards him as well?
(By Sin I am referring to any sin except disbelief in Allah)
i will answer the question shortly but in order to answer i have to correct some of your statements, firstly:
the husband is in the responsibility of his wife by having to spend on her and fulfill her needs [intercourse] whether it is sexual desire or daily life matters according to his capabilities, he also has to protect his wife from any harm that could befall her according to his capabilities, and proof for this is:
“Men are the protectors and maintainers of women, because Allaah has made one of them to excel the other, and because they spend (to support them) from their means”
[al-Nisa’ 4:34] source - https://islamqa.info/en/12465
and also:
“And of His signs is that He created for you from yourselves mates that you may find tranquillity in them; and He placed between you affection and mercy. Indeed in that are signs for a people who give thought”
[ar-Room 30:21]. source - https://islamqa.info/en/220252
When we read these verses we understand that the responsibility of a man towards his wife is blatantly different from the responsibility of a parent towards his child, and that the husband is even more responsipble for his wife than a parent for his child, but this does not mean that the husband will bear the sins of a wife if she commits a sin except if he helped or forced or did any toher thing that would make her commit that sin, period. And now to answer, No the husband will not bear the sins of the wife if she sins because of herself, but it is obligatory on the husband to correct his wife in this matter and enjoin what is right and forbid what is wrong, and if he does not, then, yes he will too, proof for this is:
You are the best nation produced [as an example] for mankind. You enjoin what is right and forbid what is wrong and believe in Allah . If only the People of the Scripture had believed, it would have been better for them. Among them are believers, but most of them are defiantly disobedient. Source - https://quran.com/3/110
and:
“Let there arise out of you a group of people inviting to all that is good (Islam), enjoining Al-Ma‘roof (i.e. Islamic Monotheism and all that Islam orders one to do) and forbidding Al-Munkar (polytheism and disbelief and all that Islam has forbidden)”
[Aal ‘Imraan 3:104] Source - https://islamqa.info/en/96371
On the authority of Abu Sa’eed al-Khudree (may Allah be pleased with him) who said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say, “Whosoever of you sees an evil, let him change it with his hand; and if he is not able to do so, then [let him change it] with his tongue; and if he is not able to do so, then with his heart — and that is the weakest of faith.” [Muslim]
and if the husband does not then the consequences are not light:
And if you turn away, He will replace you with another people; then they will not be the likes of you. Source - http://corpus.quran.com/translation.jsp?chapter=47&verse=38
Period, hence the completed answer is that no because the basic principle is that a person will not bear any of the consequences of the sins another person committed unless he did not do what is mentioned above, proof for this is:
"... It will have [the consequence of] what [good] it has gained, and it will bear... [the consequence of] what [evil] it has earned. Source - https://quran.com/2/286
