Guns are like anything else, everyone has an opinion but the truth is they're not that complicated and you don't have to spend a lot of money. It seems like I bought a No. 1 for under $400 a few years ago. It was used of course but it works great and I have every expectation that my grandchildren will get it one day.
The thing to keep in mind is the kind of game and the terrain you'll be hunting. If you hunt cornfields or prairies out west you want a heavy, long barreled rifle with a good scope. However, if you're hunting in hills, mountains or thick dense woods you'll appreciate something a little lighter and shorter with the smallest scope possible. I live in the foothills of the Appalachians so I prefer a very short barreled (18 inches in this case, its actually a little too short for the .30-06 I have it chambered in) rifle with iron sights. Out here its EXTREMELY rare to get a shot as far out as 100 yards so a scope just isn't worth it to me.
Caliber is a little more subjective. In the old days my .30-06 was considered about as powerful a round as was needed for North American game. You wanted to be careful with grizzly and buffalo and elk is an extremely tough animal, but the round would get the job done with proper shot placement. These days there are dozens of proprietary and wildcat rounds that claim to be the end all of shooting, or niche rounds that only attempt to do one thing at all. Most American hunters would never take .30-06 bear hunting anymore and some consider it overkill for whitetail deer. I like the .30-06 because it comes in so many variations you can have one rifle fulfill multiple roles, but also because its been around for a long time and is powerful and unrefined. But, it has more kick and doesn't deliver anymore power than the .308 when firing similar sized bullets. Its also considered less accurate because it requires a longer action. I could go on and on but I would be boring you. The short of it is, based on your desire for a reliable game round that is in service with NATO and so probably will always be available, you want the .308. Its very accurate, is as powerful as you need it to be and doesn't beat your shoulder up or cost too much to shoot. It is a NATO round. Its also called 7.62x54mm and its fired out of the M60 and 240. You can get almost any rifle chambered in it and it is the standard sniper round for the US military too. The M16 round (called the .223 or 5.56x45mm) is considered too small for most game. Its pretty much a varmint round, although it is an excellent round for what its designed for.
The two rifle companies too at least research (in my oh so humble opinion) are Ruger and Savage. Both make rifles on the lower end of the price scale. Ruger is known for incredibly durable, well built and acceptably accurate rifles in a plethora of calibers. Savage is considered to make the most accurate out of the box rifles on the market that are also durable and well built, although their modern design leaves the weapons looking a little industrial. I personally have a soft spot for Ruger because the company's founder was such an interesting man and because I think a weapon should be as simple and durable as possible and still fulfill its intended role. You really couldn't go wrong with either company. Winchester makes good weapons too but is a little pricey to me. Weatherby is way too pricey. Two companies I've heard good things about but am unfamiliar with personally are Saiko and CZ. They would probably be worth looking into as well. Thompson Contender also makes a really slick gun, but they only make a single shot.
Which brings us to the next question, what kind of action? Your choices are single shot, bolt action, lever action or semi-automatic. You can pretty much rule out semi-automatic right away because of price. It will be very hard to find a semi-auto .308 for under $1,500. Examples though would be the M14 (M1A1) or the AR-10. Both are fantastic weapons. A word of caution on the M14, if you look it up make sure you are looking at the M1A1 and not the M1 Garand. The Garand was the American battle rifle in WWII and shoots my beloved .30-06. Its a beautiful weapon but for two reasons, not a good survival gun. A lever action (classic cowboy rifle) can also be almost completely ruled out because the .308 is not well suited to the kind of magazine you have to use on lever actions and so is rarely chambered in it.
Where the .308 is usually found is in a bolt action. These rifles have a handle sticking out the side and after every shot you rotate the handle up and a pull it back and then push it forward to charge the rifle. These are excellent rifles. They're accurate, reliable, simple and allow quick follow up shots if one is needed.
There are two types of single shots found today. The first is the break action. This exactly like an old shotgun that hinges in the middle and you push and shell inside the back of the barrel and then close it. The Thompson Contender is an example of this kind of rifle. It is the simplest form of rifle you can buy and so tend to be the least expensive too. However, single shot rifles do not allow you take quick follow up shots. They have to be taken out of the shoulder and reloaded everytime. You have to be a good shot. An added bonus is that they are very safe.
The other form of single shot found today is the falling block. They look a little like a lever action but function completely differently. By rotating the lever forward you allow a block that covers the back of the barrel to lower and that exposes the chamber and you slide the round in, rotate the handle back and you're ready to fire. They are a little more complex (but only barely) than a break action but are also a little more elegant. The Ruger No. 1 is the only sporting rifle of this type you'll find. Although there are Sharps rifles on the market as well. They're almost always chambered for the .45-70. This round does meet any of your criteria.
Of course my suggestion to anyone is a No. 1 in either .308 or .30-06. But in reality that has no advantage over a Thompson in one of those calibers. Just my suggestions. Its so nice to feel like I know something on this board for once.
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