Cooling is a step, not a delay
The instruction to let the water cool a little casts temperature as something that happens while you wait, which quietly makes it the least controlled variable in the whole process — and the one most likely to differ between two people following the same recipe. It is worth inverting the framing: cooling is an operation with methods, each with its own speed, repeatability and equipment cost, and choosing one deliberately turns a guess into a procedure. This matters most for the teas that want the bottom of the range, where fifty-five degrees and seventy degrees are two different drinks rather than two points on a tolerance, and it matters not at all for a mug of breakfast tea, where anything off the boil is fine. The methods below are ordered roughly by how repeatable they are rather than by how traditional. The traditional ones are traditional precisely because they solve the repeatability problem without a thermometer, which is a harder problem than it sounds and one that a variable kettle solves by simply removing.
The cooling vessel, whose only job is to lose heat
The Japanese yuzamashi and the Korean sukwoo are wide, shallow, open bowls that boiling water is poured into on its way to the pot, and nothing else happens in them. The shape is the entire design: a broad surface loses heat quickly to the air, and a thin-walled vessel sitting at room temperature takes a further share into itself the instant the water arrives. What makes this better than waiting is that the drop is reproducible. The same vessel in the same room takes roughly the same amount out of the same volume every time, so a fixed routine reaches a fixed temperature without any measurement at all, and the routine can be taught to someone else. That is why these vessels exist in exactly the cultures that brew tea very cool, and why they are absent from those that do not — nobody making English breakfast tea has ever needed one. Used in a sequence, the effect compounds predictably: two transfers take out more than one, and always in the same proportion.
Transfers, and what each one costs
Every pour from one room-temperature vessel to another takes heat out, and the amount depends on the mass and material of the vessel, the volume of water, and the temperature of the room. A heavy stoneware jug takes far more than a thin glass one; a small volume loses proportionally more than a large one; the same routine in a cold kitchen in winter and a warm one in summer will not land in the same place. This sounds like a reason to distrust the method and is in fact a reason to standardise it. A routine of two or three transfers becomes repeatable by using the same vessels in the same order at the same pace, and the useful figure is not any general number but your own, measured once with a thermometer and then remembered. Practices that pour boiling water into the cups and then from the cups into the pot are doing precisely this while also warming the cups, which is one of the reasons that sequence is so widespread — it is two jobs in one motion rather than a flourish.
Waiting, and why it is the least reliable option
Water left standing in a kettle cools quickly at first and then increasingly slowly, and how quickly depends on whether the lid is closed, whether the body is insulated or single-walled, how full it is, and how cold the room is. At least two of those change from session to session in most kitchens, and none of them is usually noticed. The common instruction to wait a few minutes therefore produces a wide spread of actual temperatures, and that spread is a large part of why the same person following the same written instruction gets noticeably inconsistent tea and blames the leaf or their own attention. Waiting is not useless — it is free, needs nothing, and is perfectly adequate when the target is a broad band such as the one most black tea and oolong lives in. It should simply be recognised as the loosest of the available methods rather than as the standard one, and it should not be the method chosen for a tea that wants the bottom of the range.
Adding cold water, and the arithmetic nobody does
Splashing cold water into the pot works, cools instantly, needs no equipment, and changes the ratio at the same time — which is fine if the added volume was accounted for and a mistake if the pot was already full. It is the fastest method available and the one most likely to produce a weak cup for reasons that then get blamed on the temperature, since the drinker made two changes and only noticed one. The clean version is to make the cold addition part of the plan rather than a correction to it: decide the finished volume first, take a known part of it cold, and dose the leaf for the total. Done that way it is precise, immediate, needs nothing at all, and is genuinely easier to reproduce than waiting. It is also the only cooling method that works at the very bottom of the range without a great deal of standing about, which is why it appears in practices that brew extremely cool and want to do so quickly.
A routine worth building
Pick one sequence and keep it. Boil, pour into the cooling vessel, pour into the warmed pot, and count the seconds you take doing it, because the pace is part of the routine. Use a thermometer for two or three sessions to learn what your own sequence produces in your own kitchen, then put it away — the value of a routine is that it removes measurement rather than requiring it, and a thermometer used every time is a slower and less pleasant way of arriving at the same place. A variable-temperature kettle does the same job more directly and is worth having if the money is available. What it does not do is teach you how much heat your own equipment takes out, which is the knowledge that lets you adapt when you are brewing in a hotel room, at a friend’s house, or with a pot you have never used before.