My daughter also asked for a tubal ligation after her first child. She was 20. They also told her that they couldn't do it, not because she wasn't being rational, but because she was too young. They said and it turned out to be right that she could change her mind when her circumstances changed. That made sense to me because at 20 one hasn't had much experience, especially with babies and how not being able to have a baby could affect a future relationship.
And they were right. She married and now has a son at age 26. Again she asked for a tubal ligation and they agreed to do one after some counseling. She decided not to have it. And she's now divorced. She does have mixed feelings about having or not having it. She doesn't want any more children but she wants to still have that option if she marries again.
I'm her mother with a whole lot of experience with relationships and babies. I see that my daughter already has more than she can comfortably handle and should not have any more children. If the man that she wants to marry cannot see that another child will be too great a hardship on her then he's not the right one to marry anyway. A husband needs to put his wife's emotional health and how it will affect all children his top priority.
And so I agree that having a tubal ligation now makes more sense. You've taken care of a baby and you know by now whether or not you want more. Since you didn't mention a partner I would ask if there is one and what his wishes are. Tubal ligation is a decision to be made together if there is a partner in the relationship. Otherwise it's entirely up to you.
It is a serious decision because it cannot be reversed. It should not be made to prove the doctors wrong. Just a thought to consider. Sometimes we get caught up in our emotions and make decisions based on them instead of rational thought. I know rational is a buzz word for you. I've had many of those words myself. I married based on emotion and the thought that I had to be married (my buzz word) and I'm divorced because it was an irrational decision.