My TV is a 2016 OLED, and the tech has moved on a bit since then, but I think modern OLEDs do suffer from the same issues, just not to such a degree.
Certainly, when I was buying, OLED had three major weaknesses:
Very poor luminance performance at the black end of the brightness scale. Everyone will tell you that OLEDs has “perfect blacks”, and that’s true as far as it goes - you can turn a pixel off completely and get absolute black. But the performance just above black is very weak indeed in 2016 models. The issue is that the luminance goes up in jumps rather than smoothly: so if black is zero, then you may not be able to see any difference at all between 1, 2 and 3, then there’s a big jump to 4 & 5, etc. That leads to significant posterisation and shimmering in dark, shadowy scenes, which is hideously distracting.
Poor colour performance at the bright end of the scale (“colour volume”). OLED potentially suffers from short lifespans, particularly for the blue subpixels. To get around that, the manufacturer doesn’t use an RGB sub-pixel structure, instead it’s RGBW - each pixel contains a white sub-pixel as well as red, green and blue. At high brightness levels, a lot of the light coming from a pixel is produced by the W sub-pixel. The effect of that is that, at high luminance levels, the display becomes incapable of displaying properly saturated colours.
Poor brightness uniformity. Often, the individual pixels on the screen have different brightness levels when fed the same signal; so, when the screen should be displaying a uniform grey, instead you see lots of vertical streaks of brighter and darker greys. This varies from TV to TV, so two different TVs of the same model can vary significantly in terms of how much streaking you see. (This is referred to as the “panel lottery”). This effect is particularly visible in dark, shadowy scenes, and combines unfavourably with problem (1).
LCD (or ”LED") displays generally avoid those issues, and they can also have higher maximum brightness; but the downside is that they have poorer viewing angles, and can’t do absolute black. In cases where the screen uses Full Array Local Dimming to improve black-level, the dimming of the blacklight happens across a number of pixels, meaning that small, bright objects on a dark background can have a “halo” around them. Cheaper LCD displays tend to have poor colour saturation (although quantum-dot / QLED compensates for that).
Both types of display tend to have poor motion handling.
I’d recommend auditioning properly-set-up screens in person if you can; there’s no way to tell which set of issues you personally will find more annoying. But bear in mind that viewing a screen in a typical TV shop means A: it will not be calibrated correctly (the white balance will be pushed to blue, and contrast and colour saturation set much too high) and B: you’ll be viewing in bright conditions, so all of the black or near-black problems will be masked.