I don't really see how improved core strength would help reducing weight on hands, but I have some personal experience from some sort of connection between core strength and reduced hand problems. When I was a newbie on the bike I had problem with pressure on my hands. I stopped riding, but after a few seasons of kayaking and other sports strengthening my core I got back to cycling again, the same fit, still newbie but with much less problems with my hands. Was this due to better core strength? Maybe. I don't know.

What I do know has an effect though is that if you put out more power the core stiffens up to keep the body stable and weight on the hands are reduced as a side effect, so a general increased fitness can make you more comfortable on the bike. You can try this on the bike, ride along and feel how much pressure you have on your hands, then shift to a harder gear and start putting out significantly more power, you will notice how the weight on your hands is reduced.

Fit-wise I recommend to experiment, as sometimes what seems counter-productive has a positive effect, like lengthening the stem instead of shortening it, or lowering the handlebar instead of raising it. As you suggest moving the seat backwards is one of the first things to try though.

It's personal how much pressure we can deal with, so when someone makes a bike fit for you they can't know for sure if they put too much pressure on your hands or not. It's not just about the static pressure in a bike fit rig, but how long we ride, how hard, our style of riding etc. Bike fits always needs some longer term evaluation of the rider, so don't be afraid to adjust on your own.