PROBLEM
The non-gizmo version of the move tool has a feature where if you hold down middle mouse button, it snaps to an axis nearest to the direction of the mouse cursor movement:
It appears that this is hardcoded to middle mouse button, and can not be customized, as the Transform Modal map, which contains invidual keys for transform axis constraints (such as X, Y and Z) does not contain any middle mouse button entry:
This means that this feature for example can not work if the move tool is assigned to MMB itself. (Which is the case for Industry Compatible Keymap).
SOLUTION
Transform tool axis snap must not be hardcoded and should be present in the Transform Modal Map keymap section.