Do you handload?
The 30-06 is never a bad idea. If you handload you can come close enough to 7 mag not to worry. FWIW I'd be taking one of my 30-06's elk hunting and I have 300 and 7mm mags that would stay home. It is not recoil as much as rifle weight. I can stand the recoil. It only lasts for a fracton of a second. I carry the rifle for hundreds of hours each season. At 400-500 yards none of the magnums offer any advantage. Only when you stretch the range, and I figure I'm only good enough for 400 yards with any rifle. The trajectory at those ranges is not enough to matter on elk either. Maybe prarie dogs.
308 ain't for everybody, but it does offer some advantages if you choose your rifle wisely. There is no sense in choosing a 308 over a 30-06 in standard size and weight rifles. If you choose carefully though you can get a 308 in a rifle that is up to 3 lbs lighter and much more handy. It will give up VERY little real world performance. It'll still kill an elk at 400 yards and be a lot easier to carry up the mountains.
With a $700 budget I'd look at the TC Venture at around $400-$500. It depands on the exact version you get. They are about 7 lbs, not too heavy, not too light and are earning a reputation as one of the most accurate rifles in current production at any cost. I'd personally go 30-06, but they are offered in a wide variety of calibers including 7 mag.
I'd strongly suggest a Leupold VX-2 as the best scope for the money at about $300. That might put you slightly over budget, but I'd eat PBJ sandwiches for a month. It will be worth it in the long run.