Fitts' Law certanly does apply to touch screens.

It's about target size and distance, not screen edge limits on motion, or relative motion -vs- absolute position input devices.

The "infinite height menu bar" design uses screen edges to make the target size effectively "infinite", but it doesn't minimize the distance, and doesn't exploit any direction but "up".

FWIW, Windows does let you place the task bar on any edge of the screen though, but you still can't have four task bars, one on each screen edge!

The screen edge limits of mouse movement only apply to input devices like mice and track pads (motion sensitive), but not touch screens (position sensitive), because touch screens are position sensitive input devices, and your finger position is directly tied to the cursor position, so you can't move the cursor without moving your finger or vice-verse.

This is related to the "nulling problem" that Bill Buxton wrote about in his taxonomy of input devices, "Lexical and Pragmatic Considerations of Input Structures":

https://www.billbuxton.com/lexical.html

>One consequence of the second philosophy is that the same transducer must be made to control different functions, or parameters, at different times. This context switching introduces something known as the nulling problem. The point which we are going to make is that this problem can be completely avoided if the transducer in question is motion rather than position sensitive. Let us see why.

>Imagine that you have a sliding potentiometer which controls parameter A. Both the potentiometer and the parameter are at their minimum values. You then raise A to its maximum value by pushing up the position of the potentiometer's handle. You now want to change the value of parameter B. Before you can do so using the same potentiometer, the handle of the potentiometer must be repositioned to a position corresponding to the current value of parameter B. The necessity of having to perform this normalizing function is the nulling problem.

>Contrast the difficulty of performing the above interaction using a position-sensitive device with the ease of doing so using one which senses motion. If a thumb-wheel or a treadmill-like device was used, the moment that the transducer is connected to the parameter it can be used to "push" the value up or "pull" it down. Furthermore, the same transducer can be used to simultaneously change the value of a group of parameters, all of whose instantaneous values are different.

https://www.billbuxton.com/input04.Taxonomies.pdf

Taxonomies of Input

>Relative vs Absolute Controllers:
One of the most important characteristics of input devices is whether they sense absolute or
relative values. This has a very strong effect on the nature of the dialogues that the system can
support with any degree of fluency. As we have seen, a mouse cannot be used to digitize map
coordinates, or trace a drawing because it does not sense absolute position. Another example
taken from process control is discussed in the reading by Buxton (1986a). This is the case of
what is known as the nulling problem which is introduced when absolute transducers are used in
designs where one controller is used for different tasks at different times.