How to compare OLED and LCD brightness
What you are missing in terms of understanding is that the project writeup is not using the stock LCD backlight.  Rather it's stripping that out and instead running what is basically an &quot;LED flashlight&quot; through the bare LCD glass.  No off the shelf integral display intended for direct view is going to be bright enough for this, no matter if it's LCD+ordinary backlight or OLED.
Because LCDs are obstructive displays which darken to impede some other light source, if you can mechanically get access to both sides then you can use a bright light source up to the point where you overheat the display.
OLEDs in contrast generate their own light; as such there's no way you can increase their output beyond the point at which the tiny manufactured in organic LEDs (or else their drivers) would be damaged.  You can't use an OLED for this project.
If you want something actually workable off the shelf, it would have to be the element out of or intended for an existing projection product.  Something like that might be possible, but beware that display interconnects can be quite fragile - repurposing simple displays out of dirt cheap products where you can just start over if you break on is one thing, doing it for moderate cost products a bit more risky.