During my first month, I barely knew how to hold the scissors. You need to learn some very important basic things like preoperative assessment, positioning the patient, prepping and draping a sterile field. Some programs might have official policies regarding surgeries, but likely as you advance through the year you will gain more and more opportunity for laparoscopic and eventually open cases. What's important to gain from hysteroscopy is to be comfortable in setting up the scope and then corresponding your hand movements with the view you get from the screen developing fine precision, and then applying that to general laparoscopy. I did about the same in the beginning of my intern year. I scrubbed in with the chief on simpler cases like tubals or salpingectomies where he would show me how to place ports, introduce me to different instruments, and so on. Eventually as you become more comfortable, you'll be allowed to do more. I got to do a few laparosocpic hysterectomies by the end of the year.
Handling tissue and instruments, being certain and efficient in your movements, thinking ahead for steps is something you learn by a lot of practice. I still have a great deal of difficulty with tissue planes. Our program generally let interns scrub primary sections by the end of the first six months. That's how you become efficient at going through the motions. C/S are easy, simple procedures where you learn how to tie, how to suture, make bladder flaps, and how to be fast and good at it.
What I did have plenty of was OB and gyn clinic.
I wonder if our general surgery colleagues do more.