My understanding is that Bill English/ARC/The Augment group at SRI (Stanford Research Institute) under Doug(las) Engelbart experimented with various pointing devices, before settling on a 3 button mouse.
Some iterations had fewer (perhaps even no? I don't recall) buttons, some had as many as five buttons I seem to recall?
They even purportedly experimented with a pointing that was driven by knee movements (presumably to allow the hands to be free for other things, though perhaps this may have also been useful for accessibility much in the way there are some alternative pointing devices based upon eye tracking or breathing in more recent decades)
In SRI's studies apparently 3 buttons was considered ideal by most users?
Admittedly, they experimented with a lot of other things when it came to user input too.
For example, instead of relying solely on a QWERTY keyboard layout, NLS used a "chorded" keyboard (image attached).
Similar to playing notes on piano keys, or stenographer keyboards, multiple keys could be held simultaneously, to produce different characters.
Presumably due to the versatility of the chorded keyset (typically used by the left hand) excessive buttons on the mouse (typically used by the right hand) made it such that 3 buttons seemed sufficient?