It comes down to
1) Analog is hard, and so is high speed PWM
2) Some engineers (and online engineering resources) believe a few hundred Hz and above is imperceptible
3) It's very, very difficult to explain to someone that something is flickering, it bothers some people, but you might not be able to see it.
4) Some websites do review devices to see if they flicker. RTINGS for monitor and TV, NotebookCheck for Notebook, Tablet, Phones.
So the way LEDs, and OLEDs by extension work, is that they require the current to be limited. Usually, in electronic circuits, voltage is usually the independent variable, and current is the dependent variable, since it's affected by things like resistance, temperature change, and other factors. All displays currently commonly found use LEDS or OLEDs, whether as a backlight or for the display itself.
The problem with controlling the current, is because it "depends" on all these other factors, it has to be dynamically controlled. There's a control loop, where the circuit has to sense the current, then change the voltage, in order to keep that current constant. You could always just use a resistor, but that wastes a ton of power and cannot be used for anything more than an indicator LED. Many smaller LED drivers do this with some analog IC, but for higher power and efficiency, typically this involves switching, inductors, and capacitors. These components, to simplify a complicated matter, cause a delay in response. You can only react so fast, so you can't change the current you want too quickly, or else it will go unstable and weeble wobble. This limitation makes it hard to PWM these circuits very fast. Furthermore, it's difficult to build an analog circuit that can sense both a huge amount of current and a small amount of current accurately enough, which limits how much you can dim LEDs with this technique.
As for the second half of the problem, LEDs, particularly white ones, work by emitting blue light, and using chemical phosphors (or, more recently, quantum dots) to turn that into white light. The response of these phosphors is sometimes not very linear with respect to how much blue light comes in. If you simply dim the LED, the color might change (think, get pink or greenish).
This gives rise to the two main types of dimming used for LEDs currently: Analog (turn down the current) or PWM (turn the full current on for less time). There's also Hybrid, where you Analog dim down to 10%, and then PWM it down to however low you need, since you can't analog the current down any more.
Analog doesn't flicker, but it suffers from the LED-color-shifting problem, the can't-dim-too-far problem.
PWM suffers from the it-flickers problem, the analog-responds-slowly problem, and the it's-hard-to-turn-something-on-and-off-very-fast problem.
The problem with PWM, is that there are two dimensions - carrier frequency (how fast it's switching) and resolution (how big the increments are). Most digital circuits are limited by many factors in how fast they can switch - divide it down by how fast you want the PWM to run, and you'll see that it's often a trade off between frequency and resolution. Often, especially for something like an OLED display, where you have to actually update all of your pixels, of which there may be millions, this could be a challenge. Thus, in many designs, frequency is compromised first, because it's never advertised (but they do advertise bit depth!), and because of problem 2) and 3) above. Of course, with PWM, you STILL need the aforementioned analog feedback stuff, to varying extents, but that also limits how fast you can switch LEDs.
In the case of simpler LED displays, such as the large ones you may see in stores, the LEDs might not actually individually even have a driver, and PWM is a necessary evil due to the fact that a single driver is being rapidly switched between all of the pixels really quickly - the switching speed caps your PWM frequency.
In the case of illumination LEDs, not only do you have to deal with the aforementioned problems, but you also need to deal with the fact that inherently, the AC power grid flickers at 60hz, and to get rid of it you need capacitors and inductors. Both are expensive, and large numbers of LED light bulbs, particularly those filament kinds that are very space-limited for the electronics, flicker terribly because they omit those parts. If you look at illumination LED drivers, most quote 2kHz as the absolute-max not-to-exceed flicker frequency. Naturally, usually the limit is not what people ride, and many chips that do not support over 490Hz exist. Also, the Arduino default is 490Hz. Ugh.
In the case of phones and tablets, and OLED screens in general, some progress is being made. Xiaomi (Pocophone?) released a DC-dimming software update a while ago for some phones, which allow the user to choose whether to trade off flickering for the leds-shift-their-colors problem. However, they do warn the user that the color of the screen will be worse, and the "dirty screen effect" or poor uniformity effect will be noticeable and many customers may assume the screen was broken if this was default.
Often, users note that OLED screens flicker, even on their highest brightness setting. That is often because OLED screens actually have a hidden "Turbo" mode built in. Try to shine a bright light at the sensor on a OLED phone or tablet, and there is a chance you'll catch it "clicking" into overdrive brightness. Most phones don't let you do this manually by default because it burns the OLED screen in really quickly, but there used to be a tweak for rooted devices that let you toggle it on all the time manually.
So what can be done about it?
As mentioned in the article, DisplayMate (the publication of Raymond Soneira from the article) apparently doesn't test flicker yet. But, for monitors and TVs, RTINGS does measure that, and NotebookCheck does, for phones and tablets and laptops.
You can use those review sites to compare devices that do or do not flicker, and to what extent. If you use a device that has PWM dimming, turn the brightness up, possibly adding privacy shield screen protector to darken it if too bright. This reduces battery life, but because the LEDs are "on" longer, the "off" time will be less noticeable.
When you shop for something in a store, take your hand, stretch out your fingers in front of the screen displaying white, and then wave your hand quickly. If your fingers don't blur, but rather you see outlines, then it flickers. Same with light bulbs. Light bulbs that don't turn off immediately, but rather fade out, likely have bigger capacitors inside, which reduce flickering.
disclaimer I cannot guarantee that what is written above is correct.