I don't think that's true. Apple has been the major holdout on OLED for a while, and they're now into their second year of using it.

The vast majority of people don't have a year-old phone, never mind a flagship model.

That's true, but the Galaxy S series has used AMOLED since it was introduced in 2010.[0] Looking in more detail my guess is it's not majority market share but it's definitely significant.

[0]: https://en.wikipedia.org/wiki/Samsung_Galaxy_S_series

Granted, my 2011 Nokia N9 also had an OLED screen. But they became unfashionable for several years due to (then) poor color rendering (although their contrast, naturally, was and is top notch). Currently they've been making a comeback.