Good point. I find it very relaxing to go outside when nauseous from an infection. But not for running. For walking. The fresh air and body movement is great.

Instead of going every day 10 min you can use a scheme. Some apps help with that, but you can make your own as well (e.g. spreadsheet). You slowly build up with that via intervals, you do it every other day with one day extra rest in a week. In 12 weeks you can run 5 km straight. I do get the complexity might put one off tho.

most mild infections are viral, and I'd assume most folks with bacterial infections would not be in a place where they think about running..

You would be in real trouble if you have a bacterial infection. You will need to on some kind of antibiotic.