I think you said that you have only seen this T for about 4 times. That is pretty early for deep vulnerability. During that time, therapists often work on developing the relationship and foundation for later vulnerability. Starting with lighter stuff will give your T some insight into you and is not a wast of time. You may find that your T will slow you down if you try to go too deep too fast. In any case, talk to them about your feelings, and maybe you can come to an understanding that will be beneficial to you.
From a client perspective, I would be horrified if a T tried to force me into being too vulnerable too fast. I probably would run! But that is just me. I need more time to be certain that I can trust a T. You may be miles ahead of me in that area!