Spend a little bit more and get a 700 action, buy a barrel and have it fitted and a stock of your choice (used in the classifieds here to save $$) and for $1000-$1200 have a gun that has a very good chance if not a certainty to out shoot all the above mentioned rifles
If you are considering spending that kind of money you should definitely be building a rifle.
Here's what my semi-custom Rem 700 cost:
Donor rifle, stainless, with good laminated stock: $700
Pac-Nor super match SS barrel, fluted: $445
Action truing/chambering: $300
Shipping: $30
Trigger tuning/pillar bedding: Did them myself ($40 materials, 4 hours labor)
Total price for a budget semi-custom: $1515
This is for a gun with a heavy laminated stock and factory trigger. If you want a fiberglass stock with bedding block like the Weatherby has, and a trigger on par with the Weatherby, you can add another $200-300. So you're at $1700-1800 plus a whole bunch of leg work to find a gunsmith, locate a donor rifle, buy a trigger, stock, barrel, and maybe bottom metal. If you have a bunch of spare time to do all this legwork, and don't mind waiting several months for the work to be completed, this is a good way to go. Time = money, and the time required to build a semi-custom is significant.
You can find donor rifles for a bit cheaper than $700 but most of them are going to be blued actions with crappy stocks. Buy one of those for $450, pay $300-400 for a good stock, and another $100-150 for bluing or cerakoting the rebarreled rifle - you've spent even more than I did. You can also forego the fluting to save $125 but the Weatherby rifle is fluted. To get a magnum rifle down under 7 lbs you'd want the fluting.
You could build one for under $1200 if you are a gunsmith and don't include the price of your own labor. Since a gunsmith would not be asking about factory rifles I would say your advice in this thread is useless and inaccurate.