Of course you'll get the same effect at higher altitudes, but the difference is that you can acclimatize to them in a matter of days, whereas daily temperature changes are to high to enable the same in response to them.
I'm no expert, but considering that the body has to work harder to cool off during heat waves the effect could be non-negligible.
And there are vests that circulate chilled water [1] for military and astronaut use - although the military one linked puts the chiller in a vehicle and each passenger plugs themselves into it.
For portable use I suppose the question is whether a battery-powered cooler provides more cooling than the same weight in ice packs.
https://www.makitatools.com/products/details/DFJ201ZS
Liquid and air cooled suits are common in auto racing.
It's a bit like the "dancing man" inflatable you see outside of car dealerships. It's not airtight, but the fan keeps air coming in at the rate it flows out.
[1] https://www.theguardian.com/world/2022/aug/02/wearable-fans-...