It's funny this thread re-surfaced today. I just bought a D700 for a back up to my d500. IQ on the D600 is phenomenal, but it doesn't drive my 500 F4 very fast. Figured I would change it up since the D700 is so cheap now..
Which 500 do you have?
I know the built in body focus motor is stronger on the d700 than the d600.
But I thought your 500 was AF-S so I'm curious what you mean by it driving the 500mm?
I have the 500 F4 VR G(so yes it is AF-S). On the D500 it is wicked quick, on the D600 it is slow and sometimes just stalls. I vaguely remember the same experience when I had the Tamron 150-600, it was faster on the D7100 and D800 then it was on the D600. It might be the AF module is more sure footed? I'm not really sure.
It's not your imagination.
When I got my d750 the AF drove me nuts as I was used to my d600.
The d600 is great on subjects in lower contrast subjects, such as when I was taking pics of aircraft up about 7+miles away. This is using center focus point.
The D750, well, it just didn't like that. It wanted more contrast.
This drove me nuts for a while as I was trying to use the d750 just as I use the d600.
If i move the d750 focus point out of the center area then the d750 could focus quick on those aircraft.
Conversely, bring the subject much more to the ground and the AF on the D750 is faster than the D600. on higher contrast subjects it locks on much quicker.
this is using the Tamron 150-600 f/8 which you're familiar.
I moved up from the d7000 to the d600. I didn't notice much of a difference though my skill at that time was less than now. The d750 is a recent addition after the d600 mount got broken and I was camera-less for a bit. I tried the d500 too and loved it but needed the FX.