In the AR15, you have a maximum case length to deal with, overall is about 2.300". That means most of the larger rounds must use the AR10 action. The .22-250 is loaded out to 2.350 - which means it will be really short in an AR15.
Short loaded rounds would have to jump .050 plus the free bored leade of up to .100. Even the first .050 would cock the bullet in the barrel because the ogive doesn't have much straight section to keep it squared up. Accuracy would be horrible.
The case head diameter is .472, much larger than the AR15 bolt can safely tolerate. That also forces the .22-250 into the AR10 class, which puts it on the lower end of light varmint rounds in a 10 pound main battle rifle for full power cartridges, like the .308 and bigger.
For a varmint and crow bolt gun cartridge, it doesn't work out very well, partly the round itself, partly AR's weren't made to shoot old bolt gun rounds. The military doesn't want or need them, the guns are designed much more restrictively for just one caliber.
A read up on the 6.5Grendel and 6.8SPC will show how the AR15 mag well has affected the design of other cartridges. Both have the same obstacle in overall length, the separate designs went different ways with different goals to achieve different purposes. Comparing them shows what short or long cases, thin or fat case diameters, case head size, and powder capacity do to overall bullet length and what kind of ballistic coefficient you then get. The 22-250 is too far out of the envelope, or it would have been easy to neck down a .300 Savage case to some 6.x bullet diameter. It just won't fit.