In my primary relationship, we have decided that veto is permissible if a third party has substance problems that are creating issues in the relationship, if they do anything to endanger our children, or if they fail to disclose things that we need to know for our sexual health. Mental instability may be grounds for veto if it creates significant issues in the relationship.
Seemingly because you cannot trust your husband not to get significantly involved with a drug addict or someone who would harm you or your children. People who can trust a partner to have appropriate and healthy partner selection doesn't require this.
OP, I strongly, strongly advise that you keep away from people who are not trusted by their partners. If you think waiting 2.5 years for someone to allow you to express the way you feel about someone they happen to be in a relationship with, sure, try that method where you keep earning the privilege of using someone else's possession. That's not for me though. I want relationships where the people in them have control. Not a possessive metamour with numerous issues, all of which will negatively impact on me. To me, that's not a real relationship.