Agree with Wilkie's point, I have never been asked anthing more than name, address, d.o.b and, of late , my EA number as I'm club affiliated. There may well be a box to list medical conditions, but not having any, I don't actively look for it.
Some numbers do have contact details on the back, I think I've filled those in once, maybe twice. Its the same info thats on the entry form, plus an emergency contact number.
Unless I'm seriously injured (i.e. unable to communicate), then I should be able to make any issues known, even if I couldn't I'm not sure not having my exact details would stop a medical team responding appropriately. (If I were to collapse in the street, would the medics stand around until I was identified?)
I'm still not sure (after years of racing) exactly what race insurance covers. If I trip, walk into something, have an accident or keel over, these are just that, accidents. Not sure how (or if ) they are covered. Don't even think they should be.