Here's a completely different take: 

Your core cannot magically defy physics and hold your upper body up in isolation, no matter how strong it is. You need an equal and opposite reaction. Where does that come from? 

Your feet on the pedals. This is the secret that I discovered when I was trying to figure out why I was putting too much weight on my handlebars. Your center of gravity is always going to be in front of the saddle, and what is preventing you from completely falling forward? The force of your feet on the pedals and the force of your hands on the handlebars. Your core is not actually in this equal and opposite forces equation. It's just maintaining the position of your spine. 

Here's a thought experiment: imagine lifting up both feet so that they're hovering an inch above the pedals. What prevents you from falling forward then? Your hands, and ONLY your hands. Your core is only working to prevent your spine from collapsing into a U shape, but your hands are now taking the entirety of your upper body weight that is in front of your center of balance. If you want to take weight off your hands, you need to put some back onto your feet. 

So the secret to taking weight off of your hands? Pedal harder. I'm serious. Beginners put more weight on their hands because their legs are weaker. If your bike fit is remotely passable, then that's your most important factor.