Ok, it's not an "extra" twelve threads. The 8600k has 6 cores, that's all. The 8700k has those exact same six cores, but it also has hyper threading, when means it can process twelve threads TOTAL, six cores + six hyperthreads = twelve threads total.
Skylake, Kaby Lake and Coffee lake are all built on the 14nm Intel process, so aside from some few optimizations, support for a few different instruction sets and the die shrinks that have decreased the TDP allowing for higher clocks, they are still all basically the same. The Coffee Lake chips can be considered as the fully matured, as optimized as it's going to get for 14nm version. That being said, like I already said, there isn't all THAT much difference in the "per core" performance.
Yes, the 8600k will give you two more cores than you have now, and that's a good thing, but are those two extra cores worth 300 dollars to you? I don't really think they are. Personally, I'd recommend just waiting for the next generation. BUT, if you're NOT going to do that, THEN at least move up to the 8700k so that you now have potentially DOUBLE the number of simultaneous threads that can be processed. With hyperthreads it really doesn't work out EXACTLY that way, because there is some give and take due to parallelism, so it's not truly as fast as, say, an actual twelve core CPU would be, but since it can address twelve separate instruction processes in the pipeline, it can process much more data than six cores alone could do.
All of this is assuming that the game, or application, or operating system, is capable of instructing the processor to utilize those features. Not all of them are, and some that are, are not WELL optimized for it.
So to get to the point of the question you asked, it depends. As I said earlier, if you multitask, with recording or streaming or running separate applications WHILE you are gaming, OR, if the game you are playing is well optimized to take advantage of hyperthreading or even all of the available cores for that matter, then you will see an increase in performance.
If it is not, then you won't, and it will not be worth it. It's not a question of overkill. It's a question of whether or not it can be utilized. VT utilizes a lot of threads, usually. Not every game will be the same, even in VR.
I guess my point is that I can see a point in an upgrade to the 8700k, for the purpose of ensuring that your system remains capable enough to keep up with the demands of games that will certainly grow increasingly more complex and will likely also increasingly take advantage of higher core/thread counts, so it will take somewhat longer before your system again becomes too aged to keep up.
With the 8600k however, it seems unlikely that it will offer that same benefit for an equal distance into the future, so it is probably not AS recommendable as the 8700k. Whether ANY of these scenarios makes ANY of these upgrade options worth forking over 300 dollars for a CPU and another 150 for a motherboard, it totally up to you.