It does assume only one cutting point interior to the pizza:

"Let p be an interior point of the disk, and let n be a multiple of 4 that is greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line ⁠n/2⁠ − 1 times by an angle of ⁠2π/n⁠ radians, and slicing the disk on each of the resulting ⁠n/2⁠ lines."

My previous comment may have been ambiguous, so I've edited it to be clearer. I meant that very rarely do all of the cuts go through one point - when slices are uneven in my experience, that usually comes along with a few extra mini-triangles in the middle of the pizza where the cuts have missed each other.

I actually followed to the ham sandwich theorem article from the See Also section, and thought it was a huge coincidence that the pizza theorem article was edited a few hours ago.

Now I know the reason lol

If I'm being honest, I didn't read the entirety of the pizza theorem page, so I didn't notice that the Ham sandwich theorem was mentioned in there 😬