And we have come to the point that most of the "hair splitting" forum arguments eventually reach.
There is no argument that the .22-250 gets more velocity from any given bullet weight than the .223 can manage. The .22-250 is superior ballistically.
Whether that matters to most coyote hunters is the issue, and what "matters" is often linked to a limited number of experiential data points, ballistic mythology, and rationalization of past decisions. Once we get to that point the discussion always begins to "split hairs" and, in the worst cases, devolve into ad hominem.
I Love my .22-250 and it has never failed me when used properly. My brother-in-law Loves his .223 and it has never failed him when used properly. In our hunts together, not once has the decision about "who takes this shot" ever come down to which caliber would be used to take it. THAT is my interpretation of the OP's question. Pick one and learn to use it properly. But there is no real life reason to even have this argument. (Just like LOTS of other hair splitting arguments on this site. How about the .270 vs the .280 for an example.)
I'm going coyote hunting as soon as deer seasons end.