A better comparison might be the 6mm ARC and the 224 Valkyrie, in an AR, at gas gun pressures.
Same length barrel, limited to paper punching as the 6mm has a lot more down range energy
(
not fair 
)
Pretty much anything is going to be ballistically superior to the .223/5.56 (even the 22 Nosler

).
I don't have a 6 ARC or a 224 Valk but I'm sure someone has both and a range to shoot at some distance (1000+ yards?).
110 A-Tip in 6mm (or other equivalent high B.C. bullet) and 80 to 90 grain .224 (maybe the 90 A-Tip?)
Looking at loading data
6ARC, 110 A-Tip, 24", LVR, 28.2C gr, 2660fps, 51,200 PSI
1000yds, 35.3 MOA, 10mph wind 7.8 MOA, 1347fps @ 1000yds (energy is
468 ft-lbs)
224V, 90gr Sierra, 24", LVR, 24.5gr, 2600fps, 53,400 PSI
1000yds, 39.4 MOA, 10MPH wind 9.8 MOA, 1147fps @ 1000yds (energy is only 262 ft-lbs).
Using the same 2600fps and the 0.295 G7 of the 90 A-Tip
a calculator gets:
35.7 MOA @ 1000, 8.0 MOA, 1299fps @ 1000yds (energy is only
337 ft-lbs)
Of course the 22 Nosler, 90gr Seirra, 24" using 32.3 gr Staball 6.5, 2889fps, 53,500 PSI,
1000yds, 30.5 MOA, 10mph 8.2 MOA, 1337fps @ 1000 yds. is close (energy is only
357 ft-lbs)
Don't have published data on the .224" 88ELD, 95gr Sierra, or 90gr A-Tip, maybe in a 4DOF calculator,
But all are better than the 90 Sierra at distance, I think.
I know some will come in and say 'but I get this with mine'.
You know like 2990fps with a 90 A-Tip, 0.295 G7, SB 6.5 (22 N,
28" AR).
25.6MOA @ 1000, 6.4 MOA, 1577fps @ 1000 (energy is 497 ft-lbs)
Knocking off 100fps (2890) for a 24" barrel,
27.8 MOA @ 1000, 6.7 MOA, 1505fps @ 1000, (energy is
453 ft-lbs)
Looks like the .224 90 A-tip starting off @2890fps has close to the same thump @ 1000yds as a 6mm 110 A-Tip starting off @ 2660 .
With all those numbers I'm sure I got some of them wrong.