Strong posterior chain + Strong back + Strong grip = strong deadlift, strong snatch, strong clean.
Strong posterior chain + Strong back + Weak grip = Weak deadlift, weak snatch, weak clean.
You are only as strong as your weakest link. Intellectually this all makes sense. But grip training and hand maintenance are often times left off the training program for most trainees.
It is true that a solid program involves indirect work from all the pulling and pushing.
However every time I've introduced direct training I have experienced improved performance.
I suggest a minimalist approach. Three times per week of 3-5 sets x 3-5 reps of direct grip work at the conclusion of a training sessions. Total time invested per week will be around 20-30 minutes.
Use the same overload principles you would use to improve your squat, deadlift or press. Add resistance as the loads you are using are successfully completed.
At different times, I'll do a bunch of reps throughout the day using a constant practice or GREASE THE GROOVE approach, never reaching failure.
I also work on rolling my forearms and working finger extensions to balance out the crushing work. I carry an oversized rubber band in my pocket and toss this work in throughout the work week.
Practice direct grip work for 30 days. You will be surprised at how the core movements will become stronger.