Should Professors use COVID as Discussion Board Material?
I think I would avoid using COVID-19 as an example except in situations that are most directly relevant (those involving viral biology or epidemiology, for example), for exactly the reason your student raised.
At the same time, I wouldn't say it is inappropriate, just that it is quite a saturated topic for many people right now. It's also definitely appropriate to open forums for students to converse about managing aspects of the pandemic, especially in their experience as students, on topics like social isolation, remote learning, etc.
I would also encourage students to relate topics to COVID by their choice when there are most open-ended assignments, while also allowing them to use different cases.
Usually there is some benefit for relating content to current events to make the content more salient, but I do think this is a time where there is a bit of overload. Ordinarily, you would do this to make a bridge, where students are now thinking about the material in a relevant-to-them way. With COVID, there is enough going on that instead they might be distracted by other thoughts about COVID (including anxieties about themselves or others being infected, loved ones who are currently sick or have died, general uncertainty and impacts of measures to reduce spread, etc), unrelated to the course material, so you lose the benefits.