It would be wrong. For a start, nobody other than medical staff can say suspected anything. Having a confirmed case is dependent upon them being tested (so having to fulfil the criteria, which hardly anybody does), getting the result and then the result being communicated to the school. The conspiracy theorists and rumour spreaders wouldn't believe whatever was said. It's also dependent upon the rumour about that kid having a cough, temperature and shortness of breath being in any way true in the first place.
Giving out any information about visitors to Medical would lead to more information being demanded. The whole point is that it's private - medical is used for physical issues, for psychological issues and as a safe way of excusing people that isn't in any way connected to medical at all. Normally, nobody pays attention to somebody 'not feeling well' and going to medical - which means they can access social workers, counsellors, etc, etc, without it being remarked on. If a kid who has mental health issues or is vulnerable is now not only noticed going there, but parents are demanding to know why, they lose that safety. And that is plainly what would happen.
It also causes anxiety when the visitor could be swinging the lead. There are some who claim symptoms of whatever they've last seen on TV or think will get them out of lessons/home the quickest. I've had somebody claim they went to Venice for the weekend and they had Coronavirus symptoms this week. They really haven't (we checked) - and they didn't know what Coronavirus symptoms were, anyhow. I get kids who I listen to skipping and laughing down the corridor about how they're going home and bye for the day and then turn the corner and turn into dying swans, complete with (fake) cough as they face me. And others who have been turned away suddenly recovering thirty seconds later when they turn the corner and think I can't see them, so I can't hear them. And ones who put their heads against radiators in an effort to mimic a temperature.
I'm expecting lots of fake coughs next week. It's the Year 8 exams.
If there is a confirmed case, things change. But no, you will not be getting any personal information just because you want to know.