The one problem I see with the relay control is that when you stop
welding, the torch isn't instantly cool. Yes, it's not welding but
it's still hot. Wouldn't you want cooling whenever the torch is hot.
I have a cooling fan on my computer set up this way so that I turn off
the machine and then come by 5 minutes later or so and flick the switch
that turns off the fans. I have measured the temp of the processor
heatsink with an infrared thermometer, and if you turn everything off
at the same time, the temp spikes just after the power is shut off.

That makes sense now. The water flow behaves sort of like the argon
pre and post flood.