A desktop is more flexible, can be upgraded, expanded, can run cooler, and is easier and less expensive to repair. If you don’t need portability tower is the obvious choice.

Power consumption is highly variable in both classes, while laptops generally consume less power it is because of component selection for the purpose of battery life.

The same components and settings in a tower would consume less power as there is no load from the battery. You get some of this power back when you use the battery but not as much power as you put in. There are losses both in and out, expressed as heat.

Most running jellyfin would be looking for more 3.5" drive space than a mini PC can provide.

I have been happy with jellyfin server,

the jellyfin client for Android TV has sone hiccups. That may just be my cheap TCL TV, restarting the TV once or twice an evening gets it going again.

So far no issues with the server itself. Nor on an android phone.

I do not have thumbnail seeking enabled, we watch shows and movies through. But I have seen the option to enable such along with the warning that it will increase library discovery time.