Remember that things like screen resolution primarily add workload to the GPU and not the CPU, if you want to target a low resolution system/monitor then you'll not need a very good GPU but you'll still need a good CPU if you want to maintain good frame rates.
While there's lots of graphics settings you can turn down to lower the load on the GPU (resolution is one of them!), there's rarely any settings you can turn down to lower the load on the CPU. This means you always need a reasonably decent CPU if you want to game as it can be a limiting factor for frame rate. This is especially true if you want to push very high frame rates for gaming monitors (e.g 120-144hz), you can always turn down video settings to get your GPU to render at 120hz but if your CPU can't calculate 120 game states per second then you're not going to get 120fps and you're out of luck.
It does differ from game to game though, games like MMOs or large scale RTS or anything with large number of AI or physics tend to push the CPU quite hard, other games less so. One thing to keep in mind is that which component is the bottleneck can differ between games.