The .270Win. has been a proven performer since around 1935. Few will argue with the fact that most big game can be taken cleanly with a .270Win. Jack O'Connor wrote for years about traveling around the world and successfully hunting with rifles chambered for .270Win. Ammo can be found nearly universally. Several fine factory cartridges are marketed for that round. The options for hand loaders are nearly infinite. That said, the .270Weatherby produces a bit more on the upper end of the .270 bullet's power band and the by many reports, the .270WSM outshines the parent cartridge. So, bottom line, get any of those three and you will likely be happy with the performance. My vote would still go with the old standby, .270Win.
Given your price limit, I like the Winchester Model 70 and the Tikka White Tail Hunter. Others will like different brands. Most any rifle currently chambered for .270Win will get the job done.
The .270Win uses a .277 bullet. It is not by accident that several cartridges use the .284 bullet (7mm). Before you buy, look at all of those cartridges. Many 7mm cartridges offer more versatility then does the .270. Most impressive to me, has been the 7mmWSM. A great cartridge in a short package.