This will take each action that exists on the armature and export it with the armature with the file named as the action name. For use with Blender 2.91.

export_each_action_with_rig_as_fbx.py

import bpy
import os


filepath = bpy.data.filepath
directory = os.path.dirname(filepath)

all_action_names = [a.name for a in bpy.data.actions]

def remove_all_actions_except(ignored):
    for a in bpy.data.actions:
        if a.name == ignored:
            continue
        bpy.data.actions.remove(a)

for name in all_action_names:
    remove_all_actions_except(name)
        
    print("Exporting rig with single action: " + name)

    fbx_file_name = os.path.join( directory , name + ".fbx")
    bpy.ops.export_scene.fbx( 
        filepath = fbx_file_name,
        check_existing = False,
        object_types={'ARMATURE'},
        axis_forward = "Z",
        add_leaf_bones = False,
        bake_anim_use_nla_strips = False,
        bake_anim_force_startend_keying = False,
        embed_textures=False,
    )
    
    bpy.ops.wm.revert_mainfile()

I would love if there were a way to avoid calling revert_mainfile as it is the chokepoint of this script, especially on larger files.. I tested on ~120MB files and it took about an hour to export ~350 actions this way. Unfortunately calling 'undo' doesn't work.