L.C.
I think it is absolutely personal. There is nothing wrong feeling uncomfortable. A lot of people do.
Hotels tend to use agencies with a very good reputation. I personally used a couple in the past and always had a good experience. I did it because I had to. I had no choice.
I have to say a friend of mine years ago found one of the very best nannies she ever met (I am using her words) through the agency recommended by a Hotel in NY.