> The doctors or nurses can also test positive, so it would be EXTRA great to know the person who examined your ear infection or broken leg and was in close proximity actually ended up being sick, wouldn't it?

This right here is probably the biggest reason. The doctors and nurses are around people who might (or outright do, and are specifically at the hospital because they) have Covid, so they are at rather increased risk of infection, and subsequently passing it on to other patients. So it's good to keep contact information for patients so that if a staff member tests positive, they'll be able to inform all the patients that person saw.

And it's probably also a little bit about liability. People aren't supposed to be getting sick because they went to the hospital, so it's a good idea to be proactive about keeping track of how often that's happening. To know that, they have to be able to contact potentially exposed patients who went home to see if they test positive. If it's happening too much, they can adjust their procedures to fix it.