I have a I5-4690K at 4.3 GHz, a GTX 970, and 16GB of RAM with Stellaris on a high end SSD and still have late game lag.
I also thought about upgrading to the I7-7700K, but people using a 7700K appear to also experience lag later game as well.
Unfortunately, I don't think it will be fixed anytime soon, unless Paradox devotes a patch to just optimization. Because honestly, when a game lags using an amazing processor with a good overclock, there is a problem.
It doesn't help that so many people do not play to late game
. I see so many players that claim there is no lag, only to find out they've simply quit when they believe the game is won, not when a victory screen comes up. This wouldn't be a problem if they did this in late game, but apparently there are quite a few that simply make a new game after getting to mid game and dominating a few empires.
When I say mid game, I mean a length of time. Like, quitting the game long before they have played long enough to experience lag (which normally happens late game).
I don't know enough the I5-3570K, but if it's similar to an I5-4690K I don't think upgrading to a 7700K will do much for Stellaris.