They definitely won’t do a hysterectomy unless you have cancer or something really bad. Or maybe they would do a partial (take the uterus but leave the ovaries) because, as long as the ovaries are healthy, it is important to leave them in place for the hormones and such. Having to take hormonal pills for the rest of your life sucks! Getting the tubes tied would most likely be the best option to get done. Bilateral salpingectomy is being favored over have tubes tied or clipped these days. (Same desired effect, sterility.
A hysterectomy doesn’t always imply an ovariectomy (if it does it’s called a radical hysterectomy). You can remove the uterus without entering menopause, it’s the ovaries that produce the hormones. There could be other consequences of removing it. So I can understand most doctors would balk at a hysterectomy for birth control purposes. You’re right about doctors being wary of taking out a healthy organ. As for the age, the older you are, the less likely you are going. To regret it, so lots of doctors prefer to wait. For purely selfish reasons, they don’t want to feel responsible for hurting their patients.