Saying you want to hunt in Alaska is like saying you want to hunt in the lower 48. There are vastly different terrain types throughout the state. In some instances the 300 Win Mag might be a good choice and in other situations, I wouldn't want to be hauling the heavier weight and longer barrel that most Magnum rifles have.
The 30-06 will kill anything you need to kill in Alaska. However, the 300 WM also has some perks to it. You really need to take a look at where you will be hunting.
You mention chasing Goats. If you are planning on running up and down mountains, unless you are a 20 year old track star, you are going to want to carry as little as possible. The terrain in SE Alaska where a lot of Goat hunts take place is terribly thick with underbrush. For this type of hunt, I would be opting for a Remington 700 Mountain rifle (or something similar. I'm not sure Remington still makes the Mountain series and I can't check while on my VPN). I carry mine in a 280 Remington and love it.
A lot of SE Alaska can have terribly thick undergrowth. I like the shorter, lighter rifles for this type of hunting.
If you head up into the arctic regions, the terrain can be as flat as can be for long distances. In such an environment, maybe the flatter trajectory of a 300 WM might serve you better. As I mentioned, the extra weight and length might be an issue if you have to carry the rifle miles and miles. You can always get a lighter rifle with a shorter barrel, but your shoulder and ears might take a bit more punishment. You have to decide if that will work for you.
I love my rifles and own quite a few. I started with a 243 Win and a 30-30 Win and knew I had the capabilities of killing anything in North America except the largest of animals. I added a 308 Win and with handloads and high quality bonded bullets, I could take the largest animals as well. However, I found "excuses" to add more rifles. A 30-06 for more power. A 300 WM because it was on sale. A 300 WSM for bear but that was a bit heavy with the laminated stock, so I added a 35 Whelen for bear in the brush. A 308 Norma Mag a guy was selling at the range, etc.... So it is tough for me to pick one rifle, I always found an excuse to buy one more.
However, if I were to pick a single rifle, I would opt for a 300 WM in a light, short rifle. Recoil doesn't bother me and I've shot enough rounds in my life where I can't hear anything anyway.
For most other people, I would advise them to go with a 30-06. The Remington 700 is an excellent rifle. I own multiple Remington 700s I have to say it is my favorite platform. The 30-06 is the best balance of power, recoil, and cost effectiveness in my opinion. As I said, it will kill anything you need to kill in Alaska.
If it is worth anything, my "go to" rifle in Alaska is a 35 Whelen in a Remington 7600 with a fixed 4x scope. I have handloads with lighter bullets for deer hunting and 250 grain Kodiak bonded bullets loaded hot for times when I am treading in bear country.
BTW, I an unaware of mule deer in Alaska. The only deer I am aware of are Sitka Black Tail Deer. A big one in Alaska is the size of a small doe in the lower 48. They aren't much bigger than large dogs, so they really don't require a magnum rifle.