I would ask about cleaning procedures - that might help you feel more comfortable. The hospital I delivered at also had a fair number of policies that would not allow you to use the birth tub if you had certain infections/diseases. The baby is not really floating around in there long anyway - at least in my case brought right out and to my chest which was above the water.
I would feel certain that cleanliness and sterilization are a huge concern for any hospital or birthing center as well.
If you birthed at home you get a new tub/pool to use.
In a hospital setting (or at a birth center), the tub would be just as clean as any other surface you or the babe might touch. If you're not worried about the cleanliness of the bed, the scale, or anything else, then don't sweat the tub! And trust me, in labor, the cleanliness of that tub will be far from your mind
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I am sure the hospital has pretty strict procedures for making sure that stuff is clean.
However, I'm a little phobic about baths in general (and that has broadened into being phobic of larger bodies of water and swimming)--even if it's just from me, there's "stuff" in there. lol I can totally understand the worry. If I ever have a labor that gives me time to do anything special, I'll probably go with a nice hot shower. :)
The birthing center I had my water birth at had very strict cleaning and disinfecting procedures. Cleaning the tub after birth, from start to finish, took 3 hours. The birthing center also had swabs done, and was tested (by an independent lab) for germs and microbes. Any patient could ask to see the test results. In a tub that clean, the only germs you really have to worry about passing to your baby are your own. But birth is messy regardless of where the baby is delivered, and germs will be passed on anyway.
I recommend asking the birthing center what their cleaning procedures are, and if they ever have the tubs swabbed.