There are two main things that could potentially happen.
Relationships would in effect be a lot easier for a lot of men and women. Why? Because they would have good socials skills to avoid bad relationships and seek good ones.
Furthermore they would know when the going is good and when to immediately leave a toxic relationship so all that negativity doesn’t bring them down.
People would be happier in general because they good relationships would last and the bad ones are short lived. This doesn’t mean there won’t be any bad relationships – because sometimes it is just a compatibility issue rather than always a “bad personality” one.
Relationship skills would need to be on a new level to separate the men with the best of the best social skills from those with good ones – this is called competition theory. The top percentage of guys would be dating and in relationships with nothing changing from what it is now. Top tier men would receive attention from most women while average men would compete with each other to attract the ideal partner in an already competitive market.
If all men became better people then competition to couple with a woman with looks and a great personality would increase while competition to couple with a woman that has somewhat less than a great personality would diminish. But this is just like today’s dating and relationship market. NOTHING WOULD CHANGE. In fact, a man’s social skills when it comes to dating and relationships are continually increasing each generation (on average).
Each situation has the ability to come to fruition.