You're right that with a TC-20E III mounted, your 70-200mm will effectively become a 140-400mm f/5.6. If you use this combination at the same effective focal length and aperture as your 80-400mm (and on the same camera), you'll get the same depth of field.
You're also correct that your 70-200mm has the later generation of VR, which should give you better stabilisation, worth around 1 stop.
>AF speed with the TC-20E III on my 70-200mm VR II is >noticeably slower than the lens alone,
Not surprising with only 25% as much light hitting the autofocus sensor. One of the prime reasons for using fast lenses is faster autofocus.
Was talking to a wedding photographer a few months ago who was using the 85mm f/1.2 on her Canon and asked her why considering the price difference and reduced resolution compared to the f/1.4. While she did admit that it was a much more impressive lens to show in front of her customers, she said the biggest reason was the autofocus speed. At f/2-2.8 where she was doing most of her shooting, the sharpness was comparable. But in low light, she got faster and more accurate focus with the 1.2.
I don't use the 70-200mm / TC-20 combination much - I prefer my 300mm f/2.8 with a TC-14 - but when I do use it, AF is obviously slower but still usable. And getting back to the point of the original question, it's NOT slower to focus than the 80-400mm.
Thanks Brian for your feedback about focus speed. I've been trying to decide whether to add the TC20III to the 70-200 or to re-buy the 80-400. Seems like the TC is a practical alternative to also carrying the 80-400. Cheaper too.
Just to add a data point here, I find my copy of the 80-400mm to be sharper at full zoom than my 70-200mm VRI with the TC2e III. I do have to crop a bit (>50%) to notice, but there is definitely a difference. I did some tests both handheld + VR and on a Gitzo 1541T with a RRS BH-40 using the Nikon collars on both lenses with VR off.
Maybe I just have a good copy, but I find the IQ on the 80-400 hard to beat for the price/size.