CK Worldwide and weldcraft specify a minimum of 1 quart per minute of coolant flow to achieve the published power and duty cycle ratings for their TIG torches.
I've had a watercooled TIG torch lead failure happen before, from using an underpowered cooler pump. The coolant simply boiled away and the power cable overheated near the machine:
That happened using this pump that just sat in the bottom of a bucket. (I think it was called an "aquarium pump"; reminds me of what you posted above.) After the failure, I measured the flow and through those leads (they were 25' long), and I found the flow with that bucket pump was barely mustering .25 quart per minute. Less than 1/4 of what the manufacturer specified:
After switching it to city water pressure, the flow increased to a little over 1 quart per minute, and this extra flow was able to keep the new torch leads cool.
If you had shorter 12' long TIG torch leads, (rather than the longer 25' versions,) that would help increase the flow and cooling as well. I would advise at least measuring the flow, so you know what you've actually got.