Can use the window manager context setting operators. The property that needs changing is
scene.tool_settings.mesh_select_mode
which is a 3 member collection. All modes (True, True, True), just verts (True, False, False). To set verts alonr mesh_select_mode[0] = True.
To set them individually, set vert select mode on. Leaves faces and edge mode as is.
bpy.ops.wm.context_set_boolean(
data_path="scene.tool_settings.mesh_select_mode[0]",
value=True)
to toggle
bpy.ops.wm.context_toggle(
data_path="scene.tool_settings.mesh_select_mode[0]")
To set the mode to vert only
bpy.ops.wm.context_set_value(
data_path="scene.tool_settings.mesh_select_mode",
value="(True, False, False)")
or mesh.select_mode with no extend selection and vert type.
bpy.ops.mesh.select_mode(use_extend=False,
use_expand=False,
type='VERT')
Ok that looks good, add to 3Dview, mesh shortcuts.
Setup with wm.context_set_value
now Alt+1 will set to vert mode.