No Free Lunch
Any round that offers a significant "improvement" over the .308 will also have its own drawbacks. The same weight (or heavier) bullet at higher speeds means more recoil.
Bigger rounds also mean higher cost. Flatter trajectory is a matter of inches, literally, and is very much a question of what you can use, versus cost, recoil, etc...
In the US, the 125gr bullet is not the common deer load, We use bullets in the 150-180gr range for deer and other big game. I have had very good results with the 165gr hunting bullets, being nearly as flat shooting as the 150s, and having most of the mass advantage of the 180s.
While the .223 and the .22-250 are legal for deer in some parts of the US, in most of the country they are considered too small, rifles of .24 caliber (6mm) or larger being required. The .22-250 is about 400-600fps faster than the .223, depending on which bullet/load you are looking at.
While fast and flat shooting, the .22s have a higher wind drift than bigger bullet with higher BCs.
Also, if using the .22s (either one) bullet selection is critical. The majority of loads are varmint loads, and the bullet perfiormance that delivers exposive kills on a woodchuck is not the one you want to take deer with.
All else being equal (and it almost never is) bigger bullets tend to work better.