Hah, I was looking up the date in which CPUs first supported POPCNT and actually came across this thread. You guys are pretty funny!
It is a possibility that perhaps POPCNT is used unnecessarily, that it could be replaced by other logic, and that may or may not be true. If it is true, though, you just wont to run into the real possibility the game takes advantage of another SSE 4.2 instruction, and as such fails. The existence of an error message (instead of an outright crash) for POPCNT leads me to believe that the developers are aware of the use of SSE 4.2 instructions, and as such have placed the message instead of just having the program crash when the CPU encounters it, instead of the use of POPCNT just being some random fluke.
For those of you wondering why the game won't run without this instruction: yes, this instruction could be emulated by other instructions not present within SSE at all, but programming the game to not take advantage of that instruction set is like maintaining a game for Windows XP. There are some people on Windows XP, but maintaining the game so that it runs on XP and modern software will hurt those who use Windows 10. To make it clear, with the exception of about 4 CPUs in the Core2 lineup (out of 100? Maybe more?), all consumer Intel CPUs which do not support are, at the very least, 10 years old. You can verify this on the wikipedia page, https://en.wikipedia.org/wiki/List_of_Intel_Core_2_microprocessors. Its successor, i3/i5/i7 CPUs, all support SSE4.2 from their first generation (late 2008 for i7 release, early 2010 for i3 release, iirc).
To use an example from a recent post, the QX9650. This is an extremely high-end CPU from 2007 and while CPU improvements have slowed down in recent years, newer CPUs still have new features which can be taken advantage of which old CPUs lack.
The hard truth is that no, EA is not colluding with Intel or AMD to make you buy a new CPU, at least not just by using POPCNT. And that yes, you will have to buy a new CPU. It's very likely that most CPUs, with maybe the exception of the $1000 super-high-end 2007 CPU, are going to run at a far from playable speed even if POPCNT was not used. Likely, with more modern games coming out this is likely going to happen more and more for those who don't support POPCNT. I found a used Intel i3-550 CPU for $14.25 on Newegg (which should technically run the game, but I wouldn't recommend it), and I found the minimum spec, the i3-6300 for $168 (though instead of purchasing that because it is the minimum, I would recommend getting and Intel Core i5-9400F, or other similar newer CPU. The 9400F is actually cheaper at $160 currently). The reason Fortnite and some other games even run as well as they do on such old CPUs is probably to be attributed to the games' cartoonish stylistic theme, while Apex Legends has a much more demanding and realistic graphical style.
This thread is good reading material though.