Are you planning to stream WHILE gaming or doing other tasks like encoding and streaming on the fly, or will streaming be the only thing happening while it is doing so? If you are going to be doing heavy multitasking like streaming or encoding, ripping, or other demanding tasks, you might really want to consider a change to the Ryzen platform.
Not only would it offer better multitasking performance, it will likely be less expensive as well.
If you just want to stick with what you have now and upgrade that, then yes, it should be able to do that although it's clearly not going to do it AS WELL as a newer configuration that has more cores/threads and at least minor improvements in IPC plus support for AVX and other instruction sets that are not supported on Skylake.
Personally, I'm using a 6700K OC to 4.5Ghz and so far there has been nothing I have not still been able to do with this configuration, but it's likely I may upgrade at some point after the 9th gen CPUs are released if they look promising. Those are "supposed" to launch at the end of September, but the last Intel launch, which was Coffee lake 8th gen, happened, and then CPUs were not readily available for several months afterwards due to a lack of Intel being able to actually produce them in quantity. So it could be longer.
I think it really matters WHAT exactly you want to do with it, and what you want to invest in being able to do it.