That's not true though. AFAIK, the simpler consoles are but the more complex ones like the N64 are not. It's basically the same thing as a regular CPU. It would require a very powerful one for a software emulator to achieve cycle accuracy on more complex consoles and it would also require a larger (and way more expensive) FPGA chip to achieve cycle accuracy on those as well.
Also, cycle accuracy is not needed for a perfect re-reation. But that's another topic.
My point is, cycle accuracy is possible on both FPGAs and CPUs if you don't take cost into account.
Input latency is also not an inherent problem of the emulator itself. It's the host system that causes this. It's possible to make a bootable software emulator with zero added latency without having to rely on a separate OS. Just because nobody is interested in making one doesn't mean it's not possible.
You should read this article written by an actual developer.
https://archive.ph/2018.07.07-112551/https://byuu.org/articles/fpgas-arent-magic/
As i said, neither all FPGA cores or emulators are cycle accurate on either CPUs or FPGAs yet. But they are similarly accurate at a similar cost. LLE Saturn/PS1 can run at full speed on an i5 4670, which is a 14y old CPU. Ares can run all games full speed on my i5 12400, which i got for $150 recently. Mister FPGA, which is an affordable FPGA, can barely fit a non-cycle accurate N64 core. This core is far from perfect and pretty hacky and has maxed out the chip already.
The biggest actual benefit of FPGA is the lower power consumption and the more realistic premise of not having a bloated OS so it's much easier to achieve reduced input lag. Though, software emulators can have even lower input lag using other software tricks such as run-ahead, achieving even lower lag than a real console on a CRT if you want.