All fans turn if you blow air on them. Even if they aren't plugged into power.
But, you can always try this as well if you are handy enough and feel confident to attempt it. It's possible that there is simply too much gunk built up inside the fan bearing and no lubrication left. I wouldn't count on it but if replacing it is for some reason (Fans like this can be bought for like 5-10 bucks, sometimes less) then it might be worth at least trying.
Sleeve bearings fans have a well-deserved reputation of wearing out quickly. But they're inexpensive, and not necessarily cheaply made. They can perform very well indeed. We want to use them on our heatsinks, but we worry that they will fail and leave our CPU's overheating. Wouldn't you like to...
If the side panel is removed and the CPU cooler fan is still working, then you can probably get by without it. Seems like a lot of extra effort for a five dollar part though.