I don't personally have the TC-17E but all the MTF tests I've seen suggest that it does retain higher MTF than the TC-20E III. However, the difference is small enough that there may be a bit more subject detail with the TC-20E III as the shot is magnified more. Here is a test of the 300/2.8 with some test chart examples of the TCs about one quarter of the page from the beginning
https://photographylife.com/reviews/nikon-300mm-f2-8g-vr-iiTo my eye the TC-17E II images posted in that comparison are a bit sharper than the corresponding TC-20E III images but it might be possible to discern a bit more subject detail on the latter (look at the finest rings in the test chart).
I personally own the TC-14E III and TC-20E III and my general feeling is that the TC-20 E III gets a lot of hype but in many practical situations the autofocus performance is reduced sufficiently that it is difficult to get the best sharpness in TC-20E III images (especially in action situations and when there is some additional distance between the camera and the subject) whereas using the TC-14E III with my 70-200/2.8 FL and 200/2 II the autofocus is excellent and image quality reasonably good stopped down a bit. The TC-20E III can give acceptable image quality with these lenses on subjects such as performers in a daylight outdoor concert but I would not recommend the TC-20E III for fast action with these lenses. With the 300/4 PF I get some additional hesitation in the AF using TC-14E III and in many cases in the lighting conditions where I would use it, the 300/4 PF + TC-20 E III can only be focused using live view and not the main autofocus system used in viewfinder photography. In practice while I've obtained some good results with the TC-20E III, I rarely use it.
I don't know what kind of experience you would have with the 300/2.8 and TC-20E III if you give it a try, but I think you already have the best converter (TC-14E III) and my guess is that the larger converters probably give more grief than joy.
