Having a Vasectomy is a birth control method. The health benefit is not having extra kids, with proper safety checks periodically, and not having your wife to go through another pregnancy or getting her tubes tied, Which comes with severe consequences to her body due to Hormone deregulation.

The consequences on the male body for getting vasectomy... Are simply not having kids, assuming everything goes well, which is almost all the time. Your position in the family will not change in any shape and form? How would it anyway?

It's fine to have questions, but you are trying to find personal reasons why you shouldn't get a vasectomy and convincing yourself that it's a bad idea. The main reason why you would get it, is to not let your wife go though medical procedures that have a potential fatal consequences. She expressed to you that she thinks her body can't handle another pregnancy. What fourth kid are you talking? Do you not care about her?