I have the 70-200EFL (actually 2 of them) and often use them with either the 1.4 EIII or 1.7 EII (no EIII version of the 1.7). I can tell you the image quality with the 1.4 is superb, in fact the first time I used a 1.4 I had to look at the metadata as it didn't appear to have had any negative effects. The 1.7 is very good, IQ wise, and I don't hesitate to use it if needed. I have had a 80-400 AFS VR for many years and it doesn't get much use anymore. I have not done a side by side but my gut says that the 70-200 EFL with a 1.7 is sharper at 340mm than the 80-400 is at the same focal length. You don't mention which body you are using and that could figure into a decision. I've used the 70-200 EFL with tc's combo mostly on a D850 and that body does give you some more pixels to play with.
As for the AF, the 70-200 EFL with a 1.7 tc is faster than my 80-400. My experience, YMMV.