Do you NEED to do DIRECT core work?
Which is true regarding the training of the core muscles (abs, obliques, low back)?
A) ALL you need to develop solid abs is squats and deadlifts
B) You MUST do direct ab work for all the primary core muscles (rectus abdominis, internal/external obliques, transversus abdominis, erector spinae, QL) for core development

Which did you choose? Well, they're both wrong. As always the answer to this question is 'it depends.' Here's why.
First and foremost, training goals will dictate exercise selection. Then things like injury history, preference, or limitations in performance will also determine what exercises you choose to train and why.
The muscles of the core, the abs, obliques, and erectors, are just like any other muscle and respond to training similarly. Would you expect awesome quad development if you only ever performed wall sits? So, why would compound lifts that train the core isometrically would be enough for full development?
Let's take a look at what the research says regarding core muscle activation during a variety of exercises.

THE RECTUS ABDOMINIS (RA)
This muscle on the front of the body gives us the much desired six-pack, if you're lean enough. If you're not lean, don't worry its still in there. This muscle produces trunk flexion (sit-ups, leg raises, planks.)
Many will say that squats and deadlifts are enough to build these muscles, but our lab coat friends say otherwise. Several studies have evaluated muscle activation of the RA during compound lifts like squats and deadlifts and compared it to planks, crunches, or pikes.
RA activity was significantly higher in the 'prone bridge' aka plank exercise than any other movement tested. See chart below.

These results were duplicated in another study that used more core exercises than the plank; and a straight leg sit-up reigned supreme for RA activation. See chart below.

Here, FP = front plank, SP = side plank, SJK = a swiss ball jack-knife, and SLU = a straight-leg sit-up. Again, the front plank and straight leg sit-up outscored the compound movements for rectus abdominis activity again. *Keep in mind, the majority of this data is from EMG studies. While not perfect, EMG still seems to be the best insight we have into muscle activation from exercise.*

Rectus Abdominis: To Train or Not to Train?
It's pretty clear that the abs don't get trained much-using compound lifts. Isolation exercises to properly develop this muscle would definitely benefit most people. If you want a strong mid-section and abs that pop - think about adding in some exercises for trunk flexion like Leg Raises, Crunches, Weighted Sit-ups, etc.
But wait, look at those bars for the erector spinae activation in the back squat and overhead squat?

THE ERECTOR SPINAE (ES)
The low back muscles, the erectors, are long muscles that run the length of the spine and are responsible for straightening us back up when we are bent over. Here is one muscle group that gets trained significantly with compound movements like squats and deadlifts because the muscles are actually being loaded properly.
EMG activity for the low back extensors was VERY active in compound lifts like the
back squat, front squat, overhead squat, and isolation exercises like supermans.

In the squat, whether you train high bar, low bar, front squat, or overhead squat, the erectors are holding a constant, isometric, contraction to keep you from falling over.
The deadlift is a lot of isometric and some concentric ES contraction as there is actual hip and back extension occurring throughout the lift.
Erector Spinae: To Train or Not to Train?
It seems that compound lifts with sustained, loaded isometric contractions could be enough to train the low back erectors. This expands beyond just squats and deadlifts. Exercises like bent-over rows, cleans, snatches, and RDLs can also be enough to train the erectors.
But, what if your low back muscles are the only muscle that gets sore and fatigued after squats and deadlifts? What if every time you front squat your back rounds and you miss the lift because of your back and not your legs? If this is you - then training the erectors directly may be a good idea to build up the weak link in your lifts.
Try back extensions or reverse hypers to strengthen this muscle group or for better transfer to squats and deadlifts you can try tempo lifts where the main focus is maintaining a consistent back angle throughout the exercise.

THE OBLIQUES
The muscles on the sides of the trunk are the internal and external obliques. These muscles are responsible for dynamically producing trunk rotation, and isometrically stabilizing the trunk when forces are placed on it in the transverse or frontal plane i.e. rotation or from the sides.

Here, the side plank (SP) produced the most external oblique activity compared to any other exercise tested. Makes sense because this is the only muscle that challenges trunk stabilization in the frontal plane.
Obliques: To Train or Not to Train?
In order to develop the obliques, I think you have to add in some rotational movements and some movements that produce lateral flexion. These don't get trained too much in compound lifts because the forces on the torso are primarily in the sagittal plane. Doing single limb exercises with offset loading like split squats would require some stabilization and more activation from the obliques to maintain an upright posture throughout the lift. Isolation exercises like DB side bends, side planks, and Copenhagen lifts would do the trick for full development here.
Programming Direct Core Work
I went into A LOT of detail on this in my last article, COMPLETE CORE TRAINING, but here are the cliff notes.
Remember the principles of progressive overload and specificity should still apply with core training. Don't just do a few sets of planks once a week, but plan out some actual training like you would for any other muscle group with progressive loading and exercises to target specific tissues. This doesn't have to be over complicated. Here is a sample progression for some isometric rectus abdominis work.
Week 1: 3 x 8
Week 2: 3 x 10
Week 3: 3 x 12
Week 4: 3 x 12 with a pause in the farthest point of each rep
Here's a simple 4-week progression for some oblique work.
Week 1: 3 x 10 - 50 pounds
Week 2: 3 x 15 - 50 pounds
Week 3: 3 x 15 - 60 pounds
Week 4: 3 x 15 - 70 pounds
NOTES ON BACK PAIN
While no exercise can entirely prevent injury or pain; strength training is one of the ONLY interventions that has been shown to significantly reduce the risk of injury.
By specifically targeting the muscles around the midsection with varying stimuli you can increase your resiliency to injury by preparing to move against load and produce force in a wide variety of ways.
TL;DR Train the stuff you want to develop.
Thank you for reading!
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References
Aspe, Rodrigo R.1; Swinton, Paul A.2 Electromyographic and Kinetic Comparison of the Back Squat and Overhead Squat, Journal of Strength and Conditioning Research: October 2014 - Volume 28 - Issue 10 - p 2827-2836
doi: 10.1519/JSC.0000000000000462
Bautista, D., Durke, D., Cotter, J. A., Escobar, K. A., & Schick, E. E. (2020). A Comparison of Muscle Activation Among the Front Squat, Overhead Squat, Back Extension and Plank. International journal of exercise science, 13(1), 714–722.
Comfort, Paul; Pearson, Stephen J; Mather, David An Electromyographical Comparison of Trunk Muscle Activity During Isometric Trunk and Dynamic Strengthening Exercises, Journal of Strength and Conditioning Research: January 2011 - Volume 25 - Issue 1 - p 149-154
doi: 10.1519/JSC.0b013e3181fb412f
van den Tillaar,R. & Saeterbakken,A.(2018).Comparison of Core Muscle Activation Between a Prone Bridge and 6-RM Back Squats. Journal of Human Kinetics,62(1) 43-53. https://doi.org/10.1515/hukin-2017-0176