I can't say which is best of the states you listed, since I haven't hunted them. However here are some publications put out by the CO DOW, which give harvest success ratios by game type, unit, and season.
I would agree with jeff. Time is the biggest factor in determining success or failure of a hunt. I never forget my first elk hunt. On the next to last day of my allotted 7 days of hunting I found a real nice bull, he was a 7 x 6 and was running with a bachelor group of two other bulls one a 5 x 5 and the other a 4 x 4. well on the last day a friend and I decided to attempt a hail mary he was going to walk up one side of the hill I glassed them on the previous day while I rode around to the other side and walk up it. Hoping that one of us would find the group. As I was walking up my side of the mountain I heard him shoot. I hurried to a clearing just in time to see two elk run across the skyline some 700 yards away. Then the 5 point came over the rise and expired sliding down the slope to within 200 yards of me. Then the work started. We had to get him off the mountain because it was our last day. I remember to this day looking up the hill at that timber thinking if I only had a day or two more I'd get that 7 pointer. But, it wasn't meant to be. Funny thing was when I asked my buddy why he had shot the smaller bull his response was he shot the first bull he saw. Time constraints stink. If we had a few more days we could have hunted them in a totally different way and probably both killed bulls.
To answer your question: From your list, I would vote for Montana. Colorado has a lot of elk but they have alot of hunters too. The hardest part to hunting Montana would be getting access into the good hunting. There is a lot of forestry land up there but getting back into the really good areas requires horses and quite a bit of "know how". An outfitter would be a good investment.
"Best" is relative. Best numbers? Then Colorado. Best Quality? Arizona and Utah (No disrespect to the others) . No doubt big bulls are taken in every western state in general season units, but the real big ones are found in hard to draw units. If it is a long-term goal, start putting in for the draws and in a few years you may get lucky. I've haven't been lucky enough to draw yet, but I keep puttin in.
I've hunt a few of our fine states and I would have to go with Washington. When you apply for a tag. You pick east or west side of the mountain and hunt any area on that particular side.
Method is selected at issue of tag. The choices are archery only, muzzleloader only, modern rifle only. Although, during modern rifle you can bow or muzzleload and during muzzleloader you can bow.
There are some very large Roosevelt Elk roaming around in this state.
I have always looked for ways to save money and gain skills. I have the firearms basics covered so I rarely buy firearms these days. But I do watch for bargains. Sometimes a little TLC with simple tools will let you bring an old gun back to life without investing a lot of money. One such find was a side by side 12 gauge marked “W. Richards” It is not from the fine English gun maker Wesley Richards, it is a cheap knock off copy probably made in Belgium about a hundred years ago...