I'm torn between these processors. 10700F has 2 more cores, 11400 has better single core
That's about the size of it though. For gaming there wont be much difference, I'd be you wouldn't be able to tell if they were side by side all other things being equal. In theory the I7 should be more future proof having more cores, but by the time thats relevant its probably time to buy a new rig anyway.
The real difference will be in the rendering and streaming. From what little I know pro streamers are mostly using a separate PC with a capture card in it to take the load off of the main system. This suggests that the 10700 cores will help you more with streaming.
For rendering, what program(s) are you using? It should be fairly easy for you to Google 'Program X Intel I5/I7 11400F/10700 performance benchmarks'. Somewhere like Puget systems will have a test showing the relative performance in whatever you will be using I'm sure. My advice is to look at all you can find, not all tests have equally sound methodology but it should be obvious if most or all show similar results.
Alternatively you can ask over at somewhere like Toms Hardware forums, there might be people there with direct experience of the program you want to use. I cant speak to it directly as I dont stream or render video at all.