This happened to me. I actually suggested my T but made the assumption that she would refer the person because the person was struggling with an additction issue, and I assumed she wanted an therapist who specializes in addiction, and mine doesn't.
I tried to be okay with it, but I wasn't so I told my T about my feelings. I thought I could learn to be okay with it if we talked about it. Even though they had met 3 times, she listened to me and then made her own decision to refer the person to another therapist.

Like you, I wanted this person to get the help she wanted and I think referring her to another therapist gave her that opportunity.