Some coaches say that you don't need to train the abs directly. The argument is that if you're doing big lifts involving a lot of core work like a squat, deadlift, the Olympic lifts, push press, etc. then you don't need to do direct ab work since these exercises are heavily dependent on core strength.

Yes, ab strength is key in the big lifts, but this fact can be interpreted two different ways: 1) I don't need to work my abs since they're heavily involved in big strength lifts. Or 2) I need to train my abs hard BECAUSE they're heavily involved in the big strength lifts.

See the problem? Most smart lifters do a lot of ab work because they understand how making their abs strong can help increase their lifts. Otherwise, why are the athletes competing on these lifts (powerlifters, Olympic lifters) doing lots of ab work in their training? And if most elite powerlifters need to train their abs, what makes you think that yours are so strong that you don't need to train them directly?

And there's another issue here. In theory it's true that the big basics will strengthen the core. But that's only true if your core is functional, if you can use it properly, and if it's not a weak link to start with. But if you CAN'T use your abs properly then they won't receive much stimulation from the big lifts.

But "abs are made in the kitchen" right?

That's true to an extent. You can do all the abdominal work in the world, but if you're carrying too much fat, you won't see your abs.

Some people are gifted in the sense that if they get lean, they have good abs (visually) without training them directly. But by training your abs hard you'll make the muscle bellies thicker which will help your abdominals be more defined. Why? Because the tendinous separations marking the "6 pack" do not hypertrophy, or grow very little. So if you increase the thickness of the muscle, the separation between each part of your six-pack will be more pronounced.

Years ago a bodybuilder friend of mine did the National (US) bodybuilding championships. He finished something like third or fourth, with a very solid physique all around, great condition, but pretty much zero abs separation.

He told me he never trained his abs until a few weeks prior to the competition. Well, I told him, it's too late then. To have good abdominal separation you need to build the muscle. And that's done during the off-season, not when you're depleted and trying hard to avoid losing muscle (much less adding some). The next year he took the advice and showed up with a very good midsection.

Regardless of the goal, most people should do some ab work in their training. If you want to get stronger, be a better athlete, or just look better, it should be part of your workouts. However, there are exceptions. Just like some people don't need to directly train their biceps to make them grow, some people don't need to train their abs. There are some who have a naturally very rigid midsection and who won't benefit from ab work. But these are the exception, and your training should not be based on the exceptions.