On my phone right now so I can't post screenshots yet. But in the FBX options:

• enable only deform bones
• disable leaf bones

If you already have animations, you need to bake them, which the exporter cam do.
But I don't remember whether the FBX exporter's animation baking works well on rigs with constraints like Rigify. So try it once, and if it doesn't:

Before exporting, save as a different temporary file as we will break the eig, you need to be able to go back to the normal rig afterwards. Then hit F3 in the viewport to open the operator search bar, type Bake Action and run it. In the bake action options:

• enable clear constraints and visual keying

Then export as above.

Select the rig in Object Mode. Then in the FBX export options:

• Enable Limit To Selected Objects
• Enable Only Deform Bones
• Disable Add Leaf Bones

If you already have animations, you need to bake them, which the exporter can do. It works with Rigify. But for any scenario where it wouldn't:

Before exporting, save as a different temporary file as we will break the rig, you need to be able to go back to the normal rig afterward.

↹ Tab into Pose Mode, and select only the deform bones (in Rigify it's pretty easy, they are all in a specific bone collection so you can just hide all other collections and select all the bones you see there). 

Then hit F3 in the viewport to open the operator search bar, type Bake Action and run it. In the bake action options:

• Enable clear constraints and visual keying
• Make sure you bake in Pose data

Once it's done, you can export as the above (but disable baking at export, it's already done).