Re: What is the best .33 caliber precision long range hunting round??? Remington 700
Best will depend on what you want, and how much you want to spend. A standard Remington action will handle the ultra mags (300 rum, 338 rum, and 338 edge, which is 300 ultra necked up to 338) with normal bolt face. You can jump up to lapua bolt face to run the 338 Lapua or improved versions, including the AX. If you go up to Lapua class, you may need to go to after market bolt. I have a 338 Lapua AI built on 700 with Tubb bolt, and it is great.
By sticking with ultra mags, you might get away with using standard BLD or ADL magazine, but will likely need at least to extend magazine with Wyatt's modification to use long range bullets. Going to Lapua, you will almost certainly have to go with single stack box mag. I personally prefer the 3.9" Seekins for my Lapua and Ultra mag builds.
Brass is another issue. Ultra mags have low cost options, and brass is factory. Improved or wildcat cases require more work. Good Lapua brass is really pricey. Its even worse to buy it then fire form before you even start your loading.
If you truly want to shoot 1500 yards, you will be pushing the envelope with any gun you mentioned. They can hit accurately, but may not have velocity to open bullet. Most bullets are expected to open at 1800 fps and above. Some testing indicates you can go down to as little as 1400, but not many bullet manufacturers will agree with that (for example, Rocky Mountain bullets were tested that low with good results, but they are $2 each).
If your rifle is capable of that, it will only be that accurate if you put the right optics on it. Two pounds is normal weight of NF scope, and many of the other long range scopes are heavier.
If you want to carry the rifle, keeping in mind the added two pounds of scope, pound of bipod, etc., you may have to keep rifle lighter. Every ounce you take off of the rifle will make it that much harder to control for proper accuracy and the ability to pick up the animal after the shot.
All of that can be done. But it is a rare combo, almost the holy grail. Personally, keeping the weight of the rifle in mind, I prefer the 300 ultra mag as the best round for what you suggest, now that there are the 230 grain Bergers available. They carry extremely well to extended ranges, and can be pushed at just over 3000 fps safely. But, even at that, my rounds go below 1800 fps at around 1200-1300 fps, depending on conditions (temperature/barometric pressure/humidity).
The 338's are similarly limited, but give you a little bit of leeway on wind calculation due to higher BC of the best bullets.
Whatever you choose, keep in mind none of these rifles are magic. At 1500 yards, reading wind is extremely hard, and even a slight error in your range is a miss. For example, at the conditions last input on my Shooter ap, my fastest 300 ultra 230 grain load still drops about 27" from 1475 yard to 1500 yards. Same with my fasted 300 grain Berger load out of my 338 Lapua AI.