A teleconverter can increase your focal length, but it also reduces your exposure brightness and can reduce image quality/sharpness. You should also be aware that longer focal lengths will reduce your depth of field, making it shallower. As your 200mm becomes a 400mm, the DOF will be cut in half. To maintain a wider depth, you'd have to reduce your f-stop, but you'll already be at a loss of light by using a teleconverter. You could compensate by raising your ISO or reducing your speed, but you probably don't want to reduce speed for sports/frisbee golf.

Note: The "Nikon Z Teleconverter TC-2x" states that using it will result in 2 stops of light loss. According to B&H, it is compatible with the NIKKOR Z 70-200mm f/2.8 VR S.

Something to also keep in mind is that as you extend your focal distance, maintaining stability is going to be more difficult. Obviously the lens' 5-stops image stabilisation (assuming you are talking about this lens) and the z6ii's 5-stops of IBIS help. This is just to say that you may have to increase your speed to compensate for shake at a longer focal length, compared to what your minimum speed is now at 200mm. You can check what the stabilization rating is of the prospective Sigma lens.

As far as how effective it is on extending your zoom to a subject from the same distance, I like to review some images/videos from other photographers. Derrel Ho-Shing Photography has a youtube video that compares 200mm and 400mm for a full body shot. Wolf Amri has a different video that compares 200 and 400mm for a large building that is far away.

Actually, there are some videos comparing the use of the TC-2x and a longer lens. Definitely browse some and note the lighting conditions. Lens and gear comparison videos are a rich area on youtube.

Perhaps you could test the teleconverter by renting it from a place like BorrowLenses, LensRentals, LensProToGo, etc. I have done this a couple times with some gear.