Nothing magical about 220s, but they are probably WAAY to heavy for 308.
Long range shooting comes down to "How do I keep my bullet the most stable for the longest time" There's a lot to unpack there and lots of competing factors:
(1)"Most stable" aka get a bullet with high bc. Given a fixed diameter, higher BC generally means longer bullets. Longer bullets weigh more thus the quick "Shoot a heavy bullet"
(2) Longest time: The longer a bullet stays supersonic, the better. So shoot it FASTER. More speed = more supersonic flight. Aka pack more powder into your case.
(3) Transonic transition: All bullet design
As is evident, these two things "compete" with space in the cartridge. There are tricks of more energetic powders and loading long, but eventually your bullet gets so long that you run out of powder space. Problem being you need more powder to get that bullet up to speed.
Eventually your bullet is so large, that any gains for better BC are killed because you velocity is suffering. Larger bullets= slower velocity for a fixed charge of powder.
I like to do a lot of modeling, back of the enevelope calculations. Back a while ago, I took the known "good" 223, 6, 6.5 bullets (sorry I personnaly hate 7 and pefer 7.65 because FUDD) and scaled it up to 30 cal. Turns out the prefered bullet size would be around 200-220 grains. But here's the catch--you need to be going 2800-3000 FPS to be equivalent. What 308 Load gets a 220 Gr bullet at 2800 FPS?
There isn;t one: You need a 30 magnum of some sort
But that doesn't tell you if you can get to a mile. TLDR: Get a good ballistic 308 round 175 Federal GMM or the 185 Berger and send it. Some F-Class guy will be along to say they shoot 190s or 200s but they use longer barrels and aren't shooting to a mile.
You need Speed AND BC. Or be like me and go play with ballistic calculators.
These folks have you covered:
Load in the video or similar (to 2k yards)
Ammo hand loaded 175 grain Berger LR HPBT in Lapau brass with Federal gm primers and AR2208 powder (max safe load).
I dunno what AR2208 is, as they are Aussie, but usually its just a different name for one of ours.