I think you're using "Pixel Doubled" to mean something it literally doesn't mean.
4.5K isnt pixel-doubled anything. its 4480 x 2520. (so it would be a "doubled" 2240 x 1260... which is not a real resolution used by anyone). The only reason they are using it is because they decided they wanted higher-than-strictly-Retina PPI. Is it sharper? Sure. But it isn't a pixel-doubled anything.
5k IS a doubled resolution - its 2560x1440 doubled (5120x2880).
Pixel-doubled 1080p is UHD - 3480x2160.
All "Retina" means is "you cant see the individual pixels" - thats Apples definition.
On a 27" monitor, unless you are a hunchback who forces your face forward over your desk towards your monitor and have it pulled too close to you (which is all sorts of bad for you in lots of other ways), you're not sitting close enough to see individual pixels since you'd need to be closer than 20", and if you're sitting that close you need to back the hell up for your own physical and eye health.
Again, im not saying Apple's displays aren't "sharper" - they are, as they are the same size or smaller and have a higher PPI.
Im saying that 1440p @ 27" at the healthy distance to sit from your monitor fits Apples own definition of Retina. 4k @ 27" is WELL over Apples own definition at any distance other than "sitting unhealthily close by cramming my face into my monitor".
And 27" 4K 144hz VRR IPS displays ("retina game display") now exist for a ... fairly reasonable price at that. And that the hardware to drive them still doesn't exist so they are largely something youd buy now and sit on for another two generations of hardware to even get full use out of it.
I wont even get into the really gritty details (ive had this argument before on other forums) right now/until/unless it becomes necessary to go dig up that post and mine it for the links - but science has also pretty much proven that unless you have eyesight in the top 5-7% of the entire human race, any perceived difference beyond "i cant see the pixels" is going to be extremely slight (Yes it is sharper, can i quantify it? No, it just looks a little sharper, mostly) and unless you have those eyes, the difference is often something your brain is inventing because you expect it to be better.
Again, this is not saying it isnt sharper and isnt perceptibly sharper to people even with bad eyes, its just not enough of a difference to be quantifiable to most people beyond "that looks a little better" and in a lot of tests (LTT did a test recently and most people couldnt even tell the difference between 1440p and 4k) people will guess completely wrong as to what resolution was what.