Everything is a tradeoff. You list very desirable criteria. And they conflict a bit.
I'll agree a bolt rifle makes sense. I've built a few. My .257 Ackley with a 6x by 42mm scope and Douglas fwt bbl weighs 7 lbs with the scope.Its blind magazine in a Garret Accralite 98 Mauser stock.Its on a small ring short 98 Mexican action.I did a little bodywork.
Its been my go-to deer/antelope rifle for a long time.IMO,not to step on any toes,but if it was all I had for elk,I'd rely on sure thing precision and would probably limit myself to about 250 yds or less on low risk broadside boiler room shots..I'd likely use a 120 gr Nos partition.
I'd limit myself to 400 max under near windless conditions on deer/antelope with a 115 Ballistic tip. Nice cartridge,but it does not quite fill your bill.
I'm not suggesting the wildcats,but Warren Page made a 6.5 version of the .257AI he called the 6.5 AAR. And there is the 7x57 AI.
The only reason I mention these is that same exact rifle could be either,and the 6.5 and 7mm bullets available are better for both range and elk.
IMO,those all make sense in a Mexican 98 Mauser designed for the 7x57.
These days,no reason to use that case family (except pure preference)
IMO,I just don't pursue the belted cases anymore. You can,of course...but Kenny Jarrett pursued mostly the 280 AI for his beanfield rifles..
The game and ranges you mention match up pretty well for a 280 AI.There isn't a nickels worth of difference in the load manuals between a 7mmRem Mag and a 280 AI.
At the same time,a very good arguement can be made for getting a 270 and being done with it. (Or a 30-06)
I am quite satisfied with the accuracy I get from a #2 contour Lilja .30 bbl.
At your 400 yd criteria,the barrel will put 3 rounds on a clay pigeon size bull.
I built a fantasy 600 yd elk rifle before I decided I was not ever going to whang at 600 yd elk. I blueprinted a small ring Husky magnum action ,chambered it 30-338 and put it in a 20 oz Hi-Tec Specialties stock.
EZcarrying and 2900 fps with a 200 gr Accubond.
Whats my point? Win M-70,Rem 700,Tikka,or whatever action, in standard,30-06 length,in 270,280 (AI option) or 30-06(AI option) with a quality (you mentioned Lilja) light contour barrel free float glass bedded in a quality Kevlar/glass/foam core stock,such as a Hi-Tec will deliver.
And ,IMO,don't defeat yourself as far as light/handy with some astronomical tellescope . IMO,a 3.5-10x by 40 mm,max.
Just my opinion,worth every dime you paid for it.
Now,you can go short Magnum,Nosler Magnum, short action,titaium exotic whatever. Heck of a gun. Its your money.
PS,If you trimmed your max yardages a bit...the 7-08 is a great choice.
Some short action receivers are truly short.Others are 30-06 length blocked down. I would not choose a blocked down 30-06 size receiver to build a 7-08 for your purposes.
If I wanted 30-06 powder capacity,I would not pursue the semi-rare .284 case to get it.