I personally use a Ruger m77 markII in 7mm. I have no complaints at all about the Ruger. However with that being said I have many friends that use the Browning in both 7mm and 300 and swear by them. I do not know to many people that use tha semi-auto so I have nothing to say about those.  For myself I only use and hunt with bolt actions. So best of luck with your decision.

My elk gun is a Savage 111 7mm rem mag.  Very accurate gun and does the job on 'em.  The bull I took last year was hit right behind the front shoulder, stood there for two seconds, then fell over.  Really impressed with the gun, and you can't beat the price.  I think I paid $309 at Dick's Sporting Goods for it.

Of the choices you have listed there I would go with the Savage or the Browning.  But I must say I have a Ruger MK77 mark II in .338WM and that gun has been super accurate and very reliable...never a problem, so that would be my first chice.

not sure why you ommited the rem 700 or the win 70but  whichever i would say 300 win mag for elk...ya the 7mm is gunna do it but i like heavy bullets for heavy game. shoot a 200 grain bullet at 2800fps and your gunna break stuff when it hits.

I have never shot an Elk, but it doesnt seem like your asking about proper caliber as much as which manufactuer. I own 6 Savages in calibers from 22-250 to 300 RUM and I can tell you they are hands down the most accurate rifles Ive seen out of the box. Especially if you handload you will see the flat out accuracy of these guns. As far as Bolt vs Semi, there really isnt even a close arguement here for accuracy. The best semi's even with handloads, very rarely come close to bolt actions shooting factory ammo. SO I guess my opinion would be, get yourself a Savage in 7mm or .300 win (im partial to 7mm), practice one shot one kill and leave the semi's for the deer drives.

As for caliber you will be fine with either, just be sure the cartridge you pick has a bullet designed for big, thicker skinned, game.  For example, a Nosler Partition, Sirocco, Barnes TXS, something designed to maintain its mass as it mushrooms.

I don't care for Browning guns as "Browning" has its hands and logos on everything, boots, jackets, guns, knives, flashlights, safes, etc.  Obviously Browning doesn't actually make all those products, but merely affixes a logo...and that I dislike even more.  They can't control the quality if the products are made in China.  So I don't buy their guns.

The semi-auto vs. bolt action accuracy debate was an issue 20 years ago.  Not so now.  I have personally shot several factory produced semi-autos that outshoot my bolt guns.  The military now issues a semi-auto sniper rifle – if that doesn’t say enough about the accuracy of semi-autos, I don’t know what does.  Benelli makes a great product, though I've never shot an R-1, I wouldn't hesitate to buy one. 

Tikka is probably the best value for the money.  It's made by Sako (Finnish) and it's got a 1 MOA accuracy guarantee.  My understanding of Tikka is that their actions are all the same length - long.  Don't buy a Tikka in a short action cartridge like a 308, there is no weight savings or action length saving to be had.  Both the 300 and 7mm are long action cartridges so you will be making good use of weight and action length.

Detachable magazines are debatable.  I prefer them, and both the Benelli and Tikka have them.  I like being able to unload the gun by removing the magazine and empting the chamber.  Reloads are very easy and you don't have to have extra bullets clinking in your pocket, just a spare magazine.  For me it’s hard to unload the cartridges in an internal magazine without dropping a few rounds.  Plus on some guns, the internal magazine release is located at the front of the trigger guard, which is a great place in theory, but it hits my trigger finger in some shooting positions upon firing a strong recoiled cartridge like the 300 win mag. and causes all the rounds to dump on the ground.

Out of those I'd go with the Tikka in 7mm. But I'd prefer a Remington 700.