IF your state charges sales tax on vehicles, determine if your trade-in reduces the sales tax on the car that you are purchasing. Using your example of say 15k equity and buying another vehicle for 15k, you may pay very little sales tax, if any. It that case, it shrinks the difference between private sale and trade-in.
Do you have a CarMax close to you? If so, have them appraise your Tundra. They will give you a value that they will pay and a few days to think it over. Compare that to the time and effort (different feelings here for different people) of private sale. If you are thinking trade-in, go to the dealer that has the car you want to buy and ask them first to give you a value on the Tundra. If terrible, ask them to match CarMax or leave. If great, take it. If in the middle, compare to CarMax and decide.