First off moving to a 6C/12T CPU would help with BF1 issues to a degree but I wouldn't count on it being a silver bullet either. Yes when in servers with loads of peeps you'll likely get a boost but the places where single thread IPC is an issue (usually in single player and certain areas of maps) you won't see as much uptick though you will see some. The simple reason is the IPC difference between Intel gen 4 and 8 isn't actually all that great, especially in games at 4K. This is why people told you would not face a CPU bottleneck and it was true. BF1 is honestly a optimization nightmare. This is the other part of the problem you are having. So yeah throwing more CPU at the problem will help as it can help overpower the bad optimization but because of the fact your facing both threading issues (minor) and IPC issues (minor and arguably common even in new CPUs). Problems is both of those minor issues add up to a more serious offense and as a PC gamer myself I get your frustration. We pay a lot for hardware and expect it to perform.
If it were me I would likely do one of two things (assuming you have completely scrubbed windows and reinstalled just to be sure no software bugs/issues causing your issues). One I would either upgrade right now...or two I would decide what, in my head, is upgrade worthy. For me right now gaming at 4K with an old i7 3930K with a couple of GTX 1080s it is number 2. In my head I need PCIe 4.0 or better to drop and/or my CPU to start to bottleneck. Mainly because, on the PCIe front, SLI on PCIe 8x links for 4K gaming can cause a bottleneck. I have to choose between using one of my PCIe slots or a 5-8% FPS drop. Most games I can just deal with the drop and hold my 60 FPS but there are a few games where making sure my second GTX 1080 gets a full 16x slot (ie I disable my last PCIe last slot and therefore add in card...I guess I could remove it all together but huge pain to reinstall it again) just like the primary GTX 1080 does. I am lucky in that had both GPUs been allocated to PCIe 3.0 8x slots like mainstream CPUs are in SLI the performance hit would be even bigger, closer to 15%. So I hold for now, salivating at the prospect of when PCIe Gen 4 launches and I can buy a new CPU platform for my rig.
So I would think really hard about what you want out of an upgrade. If a few more FPS are good enough or you want to do more game streaming then jumping to a i7 8700k would probably be great. If your looking to buy another CPU platform to last as long as your gen 4 part has/can still, I might wait for the next gen AMD 3000 series parts to drop and/or Intel to finally get 10nm parts out both of which should have PCIe 4.0 as well. Even if PCIe 4.0 is not attached to their consumer launches (we know it is for Eypc 3000 series) the rumored performance uptick sounds outstanding. Big enough if you bought today....in six months time you may be sorry (amd CPU schedule wise). Yet there is always something to be said for buying what is out now in regards to PCs. Something new is always coming. It is a tough call let me know what you end up doing.
edit. To be clear I run PCIe gen 3 not gen 2 via the nvidia PCIe gen 3 patch for SB-E CPUs on capable motherboards like the Rampage IV Extreme