The short answer is: yes. The longer answer is that it's more complicated and nuanced than that. You obviously need a CPU in your gaming PC, but the component that has the biggest impact on frame rates is your graphics card. That doesn't mean you can have a weak old processor in your rig and pair it with a high-end GPU and have no worries. You still need your processor to keep your graphics card fed with data to ensure a smooth gaming experience, without that you'll get stutters as the GPU waits for the CPU to catch up.
As with everything it's all about balance. But in modern times, a mainstream CPU won't see you losing out in terms of gaming performance compared to the top chips by anything more than single digit frames per second.