https://www.cpubenchmark.net/compare/2957vs3103/Intel-Celeron-G3930-vs-Intel-i3-8100
TDP goes from 51 to 65 W, not much of an increase. If you have it set to throttle down when not used, I don’t think you’ll see much of a change.
Doubling the number of cores is a big performance boost. I would certainly upgrade if you’re considering 4K video, especially if you’re transcoding.
I’d also read the Jellyfin article about hardware acceleration: https://jellyfin.org/docs/general/administration/hardware-acceleration/
The router says it’s reachable? Weird. I wonder if it’s a bug in the router then. Any firmware update available? You have tried just rebooting the router, right?
You could also script a reconnection on the laptop to happen every six hours or something.