...I don't know how valuable the fluting is for cooling. Radial cooling fins were common for machine guns until they weren't. That may have been because they weren't very effective, or simply that the cooling benefit was outweighed by the extra cost.
Some time back I saw 2-3 references from arms boards doing testing along these lines as a way to reduce wartime costs. Think of how the Thompson evolved to be a much, much simpler gun. With: basically the same effectiveness.
The fluting usually works (some, not so much, but usually) but... enough? These reports I saw showed that it wasn't enough to matter practically. Other systems the same way, like finned aluminum cooling sleeves pressed over the barrel, which do work great on machines, electrical components, etc. I think some of the effective-value thing is that a gun works just fine over a fairly wide range of temperatures, which much other stuff does not.
(Related: the .308 conversion of the BREN did one barrel per gun. It's effectively an AR; mag fed guns can't fire fast enough to burn up a barrel, but they didn't notice for a few decades.)
In the cases of those guns that dropped cooling fins, many times the final barrel profile dropped the cooling fins entirely, so the barrel weight dropped as well.
As far as water cooled MGs today, we still have change barrel guns. It would seem plausible to make a few hundred (or a few thousand) water cooled barrels. You clip them on as needed, but don't permanently modify the gun.
I wonder if we could do something even snazzier that just open cycle water cooling. Pressurized systems with radiators instead? Ethylene glycol? Nitrate salt? Would be fun to see what could be come up with.