Dysfunctional onboard LAN controller chip getting very hot, is it safe?
There are a few reasonable ways to try and save this board


Remove the chip via desoldering
Remove the chips power supply
Damage the chip such that it won't conduct current


The easiest way to desolder the chip at home is using a heat gun. A heat gun is similar to a blow dryer but with a much higher temperature exhaust stream . If you aim the heat gun directly at the chip you may be able to lift the chip directly off the board after it heats up enough. If this device was constructed to RoHS standards, the solder requires a very high temperature before it becomes molten. Since its a BGA package, a tiny screwdriver under the corner should allow you to lift off the package if you can get the solder to melt.

Removing the power supply to just the chip would be just as good. I have no way of knowing what components power this device, but the component just above it labeled Q1LN looks like a surface mount transistor (likely a MOSFET). You could remove that and see what happens. The upside to this approach is it should be very easy to reinstall that component if the desired outcome is not reached.

The last possible solution is to simply damage the package enough that the die inside it is no longer present. Although the package is quite large, the die is likely a very small component directly in the center. Drilling directly into the center with a 1/16th drill bit would likely destroy the die entirely. However, you would absolutely have to avoid drilling through into the circuit board below. Consider this an option of last resort as this is obviously a non-reversible step.