To an extent - and that’s probably all I would focus on until you hit the benchmarks set by Chad if I were you. Most of the movement in the main lifts though is in the sagittal plane, so it won’t challenge rotational stability to the same degree as more targeted exercises. But by virtue of not collapsing under the load of a 150% body-weight squat, you’re exhibiting some degree of anti-rotational strength etc.
Jeff also has some good videos where he breaks down the major anatomical components of the core, and has some core routines that I’ve followed. That was mostly before I started cycling though, and was for strengthening my core for motorcycle racing, which is a lot more dynamic and challenging on your upper body and core - you’re moving around a lot more, under much higher g-forces, and have the added challenge of almost never being able to put weight on your handle-bars as if you’re too rigid, you could amplify any instability leading to a crash if things get out of shape. Anyway, I sort of got lazy and stopped doing the core routines (this was on top of normal strength routine). Took up cycling as good cardio to help with motorcycle racing (although I’m now way more focused on cycling funnily enough), and after a few months of unstructured, long rides, tried picking up a core strength routine again, and found I could actually do more & higher quality reps than I could before! All the bike riding is really the only explanation for the increase in core strength I can think of. So that’s another aspect to consider - spending longer in the saddle may reap the double benefit of improving your cycling and strengthening your core at the same time.
The other option is to have your strength routine incorporate unilateral exercises with a balance component as a way to get some core work in there as well. For example, my current routine involves the following alternating workouts:
- Barbell back squats
- Dumbbell bench press on an exercise ball
- Bent over single-arm dumb-bell rows
- Single-leg deadlifts with a kettlebell
- Dumbbell overhead press, alternating arms and standing on one leg
- Step-ups with dumbbells for added weight
Both of these I can smash out in under 30 minutes at the gym. Squats and deadlifts are the main lifts I care about lifting heavy for, as I’ve found the greatest benefit to my cycling there. The others, I’ve exceeded Chad’s recommendations before, and I’m comfortable continuing to practice those movements with lower weights, but in a way that promotes more balance and gives me a bit of a core workout at the same time, as well as just generally feeling healthy and what not.
All that said, if you find you have specific weak spots (again, I think Jeff has a great video on a few things you can do to try and diagnose any weaknesses in your core), it might be worth incorporating exercises targeting those areas specifically.