Oh my... I feel, that I should convert into "Cocktail-Mythbusters".
Apparently there is a "competition out there" for the coldest Martini cocktail. This is... dumb. There are bars, which are using super-chilled water, to dilute a mixture of freezer Martini premix or just gin or vodka. This is counterproductive (because of... science). Water doesn't become much colder than 0ºC. And super chilled it is max -5ºC - but doesn't really store the thermal capacity (because water has a freezing point and the "naturally coldest" temperature of 0ºC). All what you do is increasing the temperature of the drink.
Oh - there are also different ways. Point is, that these are gimmicks. If you really need to, put a bottle of premixed martini (super-super dry, vermouth has less alcohol, means, it increases the freezing point of the water-ethanol solution) into a specialty freezer, which goes below -20ºC. But as Dave Arnold have pointed it out - a too cold cocktail is no fun. It can freeze-burn your gums, it can be hazardous to your teeth. And it really doesn't "taste" at all.
So I would suggest, that a cocktail should never be colder than about -10ºC (which probably is approximates, if you take it out of a normal freezer and put into a chilled glass).
But let's talk about gimmicks in the bar... I am happy, to look for new experiences in a bar - but please don't give me any pseudo-scientific explanations. It is the controversy about "mixology" in its 2.0 expression (in the glamour days of the cocktail re-emergance around 2005-2015, some bartenders argued, that bartenders should not call themselves mixologists - and there were self-proclaimed mixologists which were not even average bartenders).
I am for pushing the bar forward - but in a reasonable and contemporary way - not in pushing a narrative, which just can be explained as pseudo-science.