Ok. You can't get a 4770k with that motherboard. The 4770k is the newest generation Intel CPU that run on the 1150 socket. The P67 motherboard you have is a 1155 socket. So if you were to upgrade the CPU alone, you could buy the 3570k or the 3770k in order to overclock. If you want a 4770k, you'll have to buy a new motherboard (not worth it).
Now to answer your question, depending on the game, the CPU matters. For example, BF4 is GPU dependent but it does require a good CPU to play BF4. The 3350p won't hold back the 780 or 780ti. However it is slower than a 3570k or 3770k. You may benefit from the higher core clock.
Knee jerk reaction, no you don't need a newer CPU. After thought, it may help.