The idea ist't completely out of reach.
in 2010, there was a cooler called "Danamics LMX Superleggera", which uses a liquid metal as a coolant and wasn't a commercial success. The alloy isn't gallium-based though, it's a sodium potassium (NaK) alloy.
NaK is way cheaper than Galinstan, but it's also highly reactive, oxidizes and may catch fire in air, and explodes when it touches water. So it can only be used in hermetrically sealed systems, and filling a water cooling loop with NaK would be outright insane.
Usually, electromagnetic pumps are used for NaK: A high current is flowing through the coolant while it is passing between two magnets. According to the right hand rule which every electrical engineer should know, a current flowing through a conductor within a magnetic field creates a mechanical force, which causes the coolant to move.
This pump principle would also work for the much safer Galinstan, but since it has a lower electrical conductivity, it will be less efficient and require more power.
I just think a Galinstan loop offers no real benefits when compared to water. With a good block and radiator design, the lower thermal conductivity of the water shouldn't be a big issue. The water is constantly mixed when passing through the block, making sure that every portion of the water comes very close to the cooling fins at some point.
It also won't look that cool, the tubes will just look like they have solid metal rods in them.
Also, considering Linus' frequent water colling spills, I'm not sure if putting a few kilograms of liquid metal in his hands would be wise .
So in short I'd say: It may work, but it's expensive and the performance may even be worse than with water. A classical example ot "Not worth it".