What PTO is the best to have when trying to have three outputs with one PTO.
I’d say sliding gears. It’s simple and compact and you can customize it to have just one output at once, all of them at once, or even 2 at a time with the way you set up your gearing. Essentially, a pneumatic cylinder would move an axle side to side, changing which gears can mesh with the one(s) on the moving axle. The different gears would go to different mechanisms on your robot.
For the sake of clarity, imagine a line drawn down the length of the moving axle, but longer than the axle. Moving the axle side to side would mean moving the axle along this imaginary line.
What do you need for the three outputs? Because if one output can be “linked” to another output (where either a certain output can activate or both can activate at the same time) any good PTO works. A common example is in backroller robots, where they have a two-output PTO and connect the intake output to the backroller with a decoupler, which is a pneumatic cylinder that can make the backroller spin with the intake, or have the intake spin by itself. The most common model for a Slide PTO has a built-in decoupler, however pneumatics can be unreliable and can break; using too many like in Slide PTOs can be risky. I recommend using a Differential (or Diffy) PTO, which utilizes two differentials and no pneumatics to independently spin two outputs. Only one pneumatic cylinder is needed for the decoupler, which for Diffy robots is outside of the PTO, near the backroller.
Of course, if you are looking to independently control three outputs, in my knowledge something like this does not exist in the VEX community (mainly because it is not really needed). However, if you pair a Diffy PTO with a highly modified Slide PTO, I think it is possible.