Hello.
My experience is, that the weight distribution of the bike for sure plays a role and a bike with 10 kg for example can feel much different than a second one with the exact same geometry, total weight of 10 kg but different distribution of the weight.
For example I feel a hitchhiker to be more stable when you have a heavier handlebar. So a light fork with heavy bars feels different here compared to a light bar and a heavy fork. I think many riders who are doing mainly frontwheel tricks want to have a light backend. And maybe vice versa.
But I don't see that it would be "perfect" to have it completely balanced, especially because I wouldn't know how one should achieve that. I mean where do you consider the middle of the bike to be? Only then you could have the same weigth in the front and the back.
So if you are really interested I suggest to experiment with it. But I guess it costs some money to buy another fork and handlebar with the same overall weight of your parts and the same geometry just to find that out... A cheap option for the hitchhiker thing for example is to put some aluminium poles or stuff like that into the grip tubes of your handlebar. If it adds around 200 g or more you can feel a difference in my opinion.
If you try something let us know!