Some games are graphics limited like fast action shooters.
Others are cpu core speed limited like strategy, sims, and mmo.
Multiplayer tends to like many threads.
Yes, stuttering is one of the symptoms of inadequate single thread core performance.
Look up the passmark performance numbers for any processor.
You will get two numbers.
The first is the total rating where all threads are fully utilized.
The second is the performance of a single thread.
As to threads, most games can not effectively utilize more than 2-3 threads,
rarely can they benefit from 4 or more unless the game is multiplayer.