Yeah, I've also got lots of experience, both as a rider and a coach, but that's irrelevant, as is yours. I'm not into saying stuff like "I must be right because I'm experienced" - I would rather rely on empirical evidence. So I'm interested to understand what evidence you have which suggests that cycling does not tax the core muscles in acordance with the demand placed upon them. I'm also interested to understand why the exercise undertaken by the core while cycling, is not enough for cycling? I'm also interested to understand your thought process behind your claim that a better core improves power delivery?
Finally, I'm still not clear what you mean by your statement "most of us have pretty poor core strength". Poor in relation to what? Can you enlarge on that?
What I mean is poor in relation to optimal. It is my opinion, that most of us, whilst we may be 'competent' cyclists, are cycling in a way which is far from optimal, and this is why we may encounter aches and pains, and injuries. I find the same with runners, or those who do not exercise.
Modern lifestyle tends to lead to poor core strength, poor in relation to what it 'should' be, and in relation to what would be optimal.
To give you a different example, let's look at the squat, which is one of the basic moves I asses in new clients. In most people it is extremely limited in range, and demonstrates massive quad dominance (look for knees tracking forwards over the toes). An elite athlete should be able to squat to very low, and will be excellent at performing a balanced squat, keeping back straight, using quads glutes, and hamstrings. A more real life example of this is in children. 'Fresh from the box', children will squat to sit. As they get older, and get used to sitting on chairs, they lose this ability. In indigenous cultures, where they do not have chairs, this doesn't happen.
So equally, the poor core strength, is by comparison to what it should be, if we were active for most of the day, rather than sitting. The relevance to the OP, is this: When riding a bike, you will do one of two things with your weight distribution. Ideally, when cycling, your 'core' i.e. your postural, inner abdominal muscles, and lower back muscles, will be doing a great job of holding your upper body weight (obviously not all). To the extent to which you have sub-optimal core strength, your weight will transfer forwards, through your arms, to your hands. This is far from ideal for bike handling, with too much forward weight bias (especially on descents), and puts enormous pressure on the hands, leading to carpal tunnel syndrome, I attach a link from British Cycling below, in which, although they don't specifically concur with my opinion, they state that it is due to 'excessive pressure' on the hands.
https://www.britishcycling.org.uk/k...he-Experts--Hand-and-wrist-pain-on-the-bike-0
Edited to add, sorry but I really don't have time, to now also explain why applying force through a lever works better from a stable platform, but hope that you can figure our from these few words the answer to your question about power delivery.