Considering the video card you'll be using is the 1060, I'd recommend considering the Ryzen 3600X. If you were doing a lot of video work to justify extra cpu grunt, vs the other points of interest you mentioned, then you could perhaps consider, say, a 3950x or even threadripper build. I can't speak to the streaming aspect, but if leaning towards the other points of interest e.g. VR and gaming, the 1060 would seem to be the weak point rather than a value cpu. Your consideration of going from your 1060 to 1660 or 2060 may be more of a side-grade than an upgrade to be worth it.
With a flexible budget, in hindsight, it may have made more sense to allocate a bit more on the video card front. That said, it's certainly not like you're in a bad spot. A 3600X should do you proud for some time (perhaps paired with a video card upgrade in the not too long-while future should you so wish, since I'd suspect down the road you'd be held back by the 1060 well before the 3600X).
So in a nutshell, my recommendation would be not to unnecessarily cpu splurge given your planned usage and simply get the 3600X since that sounds like it should serve your needs nicely and would provide great bang for buck. Perhaps the 3800X should you feel inclined to splurge a bit. The money you might have otherwise spent on a high-end cpu could be slated for the next build -- the AM5 platform, DDR5, et al. is not all that distant away.