@Multipack
Yes, I've seen some of your prior posts, I fully understand you have somewhat of a technical background, and yes that's easy to give recognition and credit for.
I presumed OP wasn't going to be upgrading the entire PC, just a cpu upgrade, since that was discussed, a move to a 4960k etc so was taken in that context. Many OP's don't know that you just can't add a cpu, you must remain in the family, so wasn't aimed specifically at you.
IPC of sandies isn't that far below anything. The 2500 is the best cpu for that H61 board apart from the 2600, which just exasperates OP's issue. Heat. The bonus to sandies is their soldered lid. Granted the 3570k has better IPC and higher clocks, but the difference being the TIM used on Ivys makes them more suspect to heat issues, the sandies handling heat far better.
As far as TJ thermals goes, case heat has a lot to do with them. In climate controlled area, not so much, but in open air applications, ambient temp inside a case when used in conjunction with an air cooler can play hell with cpu temps. I'm assuming that you know that its impossible to lower cpu temps to below ambient by mechanical means, so if case temps are in the 50's, the minimum temp at idle will be in the 50's, closer to 60's. Add in gpu temps under gaming conditions and air cooled cpu exhaust, case temps will be in the 70's, which greatly reduces the air coolers ability and leaves cpu temps closer to 80's and getting hotter by the minute. Gotta have airflow. Its an absolute must or the heat becomes cyclic , constantly feeding continuously hotter air, further reducing the cpu coolers ability until after a while it reaches throttling temps.
If you think background processes are unimportant, you are mistaken. not only do they absorb processing power, but they do not allow the pc to idle. OP has enough to handle with lower IPC from his sandie, no need to kill any source of performance. 20% background can take my 34 degree idle 3570k to 55 degree, and that's liquid cooled. Shutting down things like Indexing, antivirus checks, google updates, win10 searches etc and regulating them to idle only times, starting after 15mins of idle, and not allowing them to run during gaming will only help, it frees up processor usage, a cpu running 70% runs cooler than the same cpu at 90%. but that's not news to you, but may be to OP.
You are perfectly correct in your suggestion that a better cooler will make a world of difference. But only in the sence that the temps will be temporary and subject to case ambient temps. Even a Noctua NH-D15 can't do much when case temps are in the 50's, it'll still see 50's and better, but for a short time will have much lower temps than the stock cooler, until the cpu cooler itself becomes acclimatized, in which case it'll still have high temps, but is better able to absorb them.
OP needs airflow. With just a single exhaust fan in a hot ambient, his case temps are killing performance, throttling the cpu. Getting that heat out of the case is just as important as keeping the cpu temps down, they go hand in hand.
I've been inside PC's for over 30 years, have an IT engineering degree and a decent background in Thermal Dynamics. Give me some credit too. You are talking about performance as it applies to FPS. I was talking about performance as it applies to being actually able to get the FPS by maintaining the cpu in nominal operating temps. Slightly different direction, even if the destination is the same.