I mean, there are practical and non-nefarious reasons not to tell people that your Commander in Chief is out of commission.
If it goes badly he will be dead. You can’t hide that. If it goes well and he recovers in a few weeks he will still need to be quarantined for those few weeks. So you wouldn’t really see him or anyone around him so people will automatically assume he has contracted it. Especially since he was known to have been hanging out with several people who have now tested positive.
So lying about the Presidents health may be a good idea in certain situations. It almost certainly isn’t in this specific situation.
Edit: Also, if he lies about it, the hundred of people who have been around him might not take the actions they need to and continue making things worse fornthemselves and those around them.
Trump has taken MANY actions that defy conventional logic about Gain/Loss and Future Predictions.
I think a lot of people view the his primary job to be inspiring confidence in our leadership, and avoiding hysteria and mass panic over the virus. If the person who is supposed to be telling us everything is ok, is actually not ok themselves, the panic could grow very quickly.
I agree with you that it would be stupid to lie. But that doesn't mean they wouldn't be tempted to do it anyway. Especially because we could just have messages about everyone being quarantined for safety and they could get away with hiding the President for awhile.