There are two reasons for this trend - recoil and wind drift.
New bullets are making smaller lighter bullets equal ballistically to the larger bore/heavier bullets.
Today, 6mm bullets are quickly approaching the vaulted 6.5's and the 7mm have met/exceeded most 30cals. Only the 300gr 338 MK stands alone unless of course we get new bullets from Canadian suppliers.
So from a competitive stand point in weight restricted rifles, why put up with excessive recoil AND barrel heat. The less powder you burn, the lower the barrel heat per shot, the better your groups are likely to be over 5 to 15 shot strings.
Having just shot to 1000m in some nasty winds, I can say that the 162gr Amax can hold its own very well. If driven half decently, actually win.
Since this is paper punching, there is little need to worry about impact energies, bullet expansion, etc. Why not use the lightest bullet to do the job ESPECIALLY if case size is restricted.
I own a 220gr MK launching 300RUM (3000fps) and a 7 Mystic launching 162gr AMax at 2900fps (I can easily reach 2950fps). At 1000m, the doping for either is very close and if you compare ballistics you will see why.
Both have essentially the same BC and similar velocity. The Sierra numbers tend be right on and the Hornady tend to be conservative. Essentially the same.
If you are limited to a '06 case, even when improved, you are going to be hard pressed to push a 200gr MK, let alone anything heavier, much over 2800fps in a very long barrel. Odds are you will max out around 2700+fps as best accuracy is rarely ever at max/over max pressures.
For reference, I compared the load data for the 300RSAUM which is pretty much the same case volume as an improved '06 like the 30 Gibbs. They list 200gr loads going 2700-2800fps in 26" barrels. Let's say you gain 50fps by going a bit longer so really you will be shy of 2900fps.
With the 220gr MK and a very long throat, I would really be surprised if you could safely exceed 2700fps. A McPhee barreled 30-338 at the shoot was around 2800fps and very high pressures with the 220gr MK (bolt tough to open - not a good sign).
You are still ballistically well behind an improved 280 like my 7 Mystic pushing the 162gr Amax. Now add the 180gr Bergers to the mix and you are pushing a bullet equal to the wonderful 240gr MK at typical big magnum velocities (2800fps). Do the wind drift comparison on that.
If you want to reduce time of flight, you could build a 6.5-06 and push those 142gr/140/139gr match bullets to 3100fps. Since all have BC equal or higher then the 200 (0.565) to 220gr MK (0.625), there is a huge advantage when the wind blows.
Experiment to your heart delights. I know I do and since I already have played with most every calibre at long range, I can offer the advice that I do.
For the Osoyoos location with its wacky winds, I would consider an extremely fast 7mm as the best choice. A 7STW pushing 3300 to 3400fps with that Amax, 3100 to 3200fps with the Berger would be an ideal wind bucker. However, wouldn't choose it for a high shot string match or long barrel life.
In a world with unlimited budgets, a long barreled 338-408Chey Tac pushing 300gr MK (0.8 BC approx) at 3300-3400fps would be pretty hard to beat for wind drift.
Either that or learn how to drive a lot better.
Jerry