Not as much as you think. In the situation you described, you need a knife to:
1. Break down wood
2. Start a fire (I'm assuming you know how to make a bow drill and how to make strong enough cordage with nothing but a knife...)
3. Find water? How?
4. Set traps or make tools to attain food (again, given that you know how to make strong enough cordage out of yuca/bark/grass/etc.)
Non-knife solutions:
1. Set the wood against a rock or stump so that the section you want to break is off the ground. Throw a rock onto it or break it with your foot. If your smart you'd just find the right size wood to start with
2. Shoulda brought a flint and steel. That or you could easily find the pieces to make a bow drill; scrape them on rocks to shape them. Get cordage using the same method you would have. (Use a sharp fragment of a rock instead of a blade.)
3. While you're looking for the right size wood for a fire (see 1), look for water also. How does a knife help you find water?
4. Most traps don't need much if any cutting of branches. Set a snare using the cordage you made in step 2 and a springy limb.
See? I would happily trade a knife for a fire steel or a water filter in a survival situation. Better yet: Be prepared, and have the right gear from the get go, and don't need to "survive."
Again, i have what equipment you listed and a whole lot more. The Knife was backup. Also, Im confused. If you can do all that without a knife, then how can having a knife, simply giving you even more options be useless? In survival options are good. I dont want to argue. We should be friends. I get that a large group of people are fixed up on the end all meets all master knife. Im aware (as stated) of its lack of existence. Hence my survival kits(s) for different uses. You seem to have me confused. I dont want to argue. Its was a simple (yet high requirement) question. I should not be hated and flamed for my inquisition. It was not if you could have any 1 tool, it was "if you can have any 1 knife".