To support yourself upright you use your glutes. This is why when you climb you tend to lean over, to recruit your glutes. They're the most powerful muscles you have and you want to use them when riding if possible. I've found myself finishing races with really sore glutes, indicating to me that I wasn't using my glutes in training. Therefore when I train I consciously spend a significant amount of time on the drops or in a lower position if on the hoods/tops.

Once you're conditioned to the point where you can use your glutes a lot you won't put as much weight on the bars. It requires cycling fitness, not necessarily core fitness.

Try this - while leaned over just take your hands off the bars. Try it on the trainer or outside if it's safe. You'll automatically start to accelerate as you try and support your weight without using your arms. If it was just core you'd maintain speed but you end up using your glutes to keep your hips from rotating forward so you'll accelerate.

Once you're "cycling fit" you'll find that you can climb hills while loosely holding the bars with less (but not "no") weight on the bars. This is because you're naturally putting down a lot of power through your glutes, something a new or lessor trained cyclist can't do.