First I'm hearing of this. Last I checked, air coolers had basically reached parity with any lower-end water cooled setup.
My guess is manufacturers don't want to tell people they should air cool if it requires listing specific models. It's easy to just say they recommend water cooling since basically all water coolers will provide adequate performance.
That said, I think liquid cooling has reached critical mass. AIOs are commonplace.
I think it would be (uh) cool to have a extra huge external reservoir and fan (think motorcycle or car radiator plus maybe a tank) that could be nearly silent and cool the cpu and gpu.
Despite the fact that I think that it is very likely that a $40 cooler like the one mentioned by you would work well enough, when I will build a new computer with a top model AMD Ryzen CPU, which dissipates up to 200 W in steady state conditions, I will certainly buy a Noctua cooler for it. A computer with an Intel Arrow Lake S CPU would be even more demanding, as those can dissipate much more than 250 W in steady state conditions.
The reason is that by now I have the experience with many Noctua coolers that have been working for 10 years or more, even 24/7, with perfect reliability and ensuring low noise and low temperatures.
I am not willing to take the risk of experimenting with a replacement, so for my peace of mind I prefer the proven solutions, both for coolers and for power supply units (for the latter I use Seasonic).
Noctua knows that many customers think like this, so they charge accordingly.
But the noctua fans are reliable, but really quiet.
Your ears are worth it.
It’s still gonna be louder than a water cooler if that’s your primary concerns. Otherwise other air coolers are only marginally less silent and just as effective at half (if not less) the price.
Specifically the pumps. Those things have an obnoxious high-pitched whine that I personally find unbearable, especially during low/idle workloads.
It's possible that the actal dB level is lower, but the frequency and sound characteristics matters. A lot.
*Technically the truth
Sometimes the solution is worse than the problem. My favorite example is the TRS-80 Model II and its descendants, with the combination of the fan and disk drives so loud that users experience physical discomfort. <https://archive.org/details/80-microcomputing-magazine-1983-...>
- Inner voice: "You don't miss the old PC noises, you just miss those times".
- Shut up!
But this only simulates keyboard and mouse click sounds. In any case, you wrote "whenever you start a game or app" (my emphasis). The Model II's fan and drive noises are 100% present from start to finish, with the combination enough to drive users insane (or, at least, not want to use the $5-10,000 computer).
Some coolers today still look like that but they're on chips drawing 35W or so while idling at <2W.
[1]: https://www.cpubenchmark.net/cpu.php?cpu=Apple+A18+Pro&id=62...
[2]: https://www.cpubenchmark.net/cpu.php?cpu=Intel+Pentium+4+3.7...
Even worse than the TDP was the fact that the 90 nm Pentium 4 had huge leakage current, so its idle power consumption was about half of the maximum power consumption, e.g. in the range 50 to 60 W for the CPU alone.
Moreover, at that time (2004) the cooler makers were not prepared for such a jump in the idle power consumption and maximum power consumption, so the only coolers available for Pentium 4 were extremely noisy when used with 90 nm Pentium 4 CPUs.
I remember when at the company where I worked, where we had a great number of older Pentium 4 CPUs, which were acceptable, we got a few upgrades with new Prescott Pentium 4. The noise, even when the computers were completely idle, was tremendous. We could not stand it, so we have returned the computers to the vendor.