The .270 is a great cartridge! Buy one, and you'll love it!
Not that long ago it was considered more than adequate for anything that walked in North America. Then other cartridges became popular, and now it's deemed by some to be too weak for long shots, or for large game.
In reality, all you need to do is poke a little hole in an animal in the right place, and it's gonna die quite quickly. Accuracy kills animals. Power is secondary. To be fair, power can make up for a slight deficiency in accuracy (and maybe let you try again). Then again, accuracy can make up for a large deficiency in power...
I've owned and shot loads of rifles and lots of calibers, and the .270 is my hands-down favorite (although the 22-250 is a close second)! It's an easily available, affordable cartridge, with gobs of possibilities when it comes to reloading (just like the 30-06).
The difference between a .270 and a .30-06 is marginal at most ranges. The .270 goes a bit quicker, and flies a bit flatter, which helps if you're not spot-on with range estimation. The 30-06 hits a bit harder, but not enough to really matter, at least on anything near realistic ranges. In the end, it all comes down to that accurately placed little hole... Side-by-side, it comes down to personal preference, the difference is not enough to really matter for the vast majority of shots. Actually, I can't think of a single instance where it would matter. But, maybe...
I shoot my rifles out to 550 yards on the range (actually, behind the house). Although I've moved past that point in my life, I've reached way out and made clean kills on mule deer, whitetail, and antelope at ranges I'd never dream of shooting anymore. I made a clean, quick, one-shot (low-lung and top of heart) shot on a mule deer buck at over 600yds, and antelope at over 450 yards with the .270. Whitetails at 300-350yds are no problem at all. Personally, if I was going to hunt bear, elk, or buffalo with a center-fire, I'd do it with the .270 but I'd get inside 200 yards and wait for a good, clean, broadside or quartering-away shot. I wouldn't necessarily use the same BULLET for those animals, but I'd use the same rifle... The .270 doesn't lack for power in reality. On paper, it has less power than the magnums, which might matter if you hunt elephants or similarly-sized critters. I'd argue instead that the Magnums are ridiculous for most North American animals. But some guys like 'em, so let 'em have 'em. To each his own. Animals shot with a xxxMag aren't any deader than they are with a .270.
Again, I wouldn't make those long shots anymore, mainly because I don't find I'm interested in the long-range facets of hunting anymore. I prefer the (different) challenge of close-range hunting with flintlocks rifles and longbows. I'm also more experienced, and more concerned with the things that can go wrong on those long shots. There's plenty of time for a standing, stationary animal to simply take a step after you've pulled the trigger, turning a great shot into a horrible mess. Or a gust of wind, or almost anything else... There was a time though, when I thought getting something, anything at all was the most important facet. I also thought long shots were fun, rewarding, and challenging. They are, but paper (or steel) works fine for that. Now, I use the flintlock to hunt, and the .270 to grocery shop.
If you're looking for accuracy, I'd go for a bolt action, and I wouldn't be put off by a Savage (my bro-in-law owns several, and likes them). I like Winchester and Remington, but the deciding factor is going to be YOU, not the rifle. Matching the bullet to the rifle is important, and a good scope is a must. If money is a factor, spend less on the rifle, spend more on the scope. Realistically, you won't be as accurate as the rifle is, unless you're a well-practiced marksman. Even then, a good shot can make almost any rifle look good, where a great rifle in poor hands will appear to shoot terrible.