Why would Putin want people to think that Putin is sick or infirm, and his ability to lead might be compromised? His entire schtick is that he's a strongman leader and the only reliable option in Russia.

It's basically targeted maketing. They pretty much wanted the West lethargic and unreactive at some point early in the war. Like "don't bother sending aid to Ukraine because Putin is kicking the bucket soon" kind of a narrative.

In Russia yes, they project the strongman narrative.