Mine are older now, but all are seasoned and keen cyclists.
I bought fairly standard 16", 20" and 24" MTB look-alike bikes from my LBS. The smaller ones were quite heavy, although the 24" were light.
Much is said about the weight issue for bikes for very young children and I find it a matter of almost no consequence.
Geometry is key. Quality and rigidity are important. Fit is an issue. A few kilos here and there are neither here nor there. Because children don't weigh 70+ Kg, their bikes will always weigh more proportionally. It has ever been so.
How does a slightly heavier bike set a child's cycling back by a year, as Ian Cooper claims above? A moment's thought begins to suggest it's bunkum.
A child of five, six or seven will not usually be going up monster climbs. If they are, it will not be done competitively.
My middle child rode a heavy, 5-speed, 20" bike to his grandparents' house 70 miles away, aged 8. It was his idea and he loved it. He dealt well with big hills going round Brecon. They put a banner out for him at the farm when he rode in. He's 16 now and still rides whenever he can (does those 70 miles in under 4.5 hours at a canter now). Neither he nor any of my children has ever mentioned bike weight being an issue. I didn't notice it being one. None of my cycling friends has mentioned bike weight being an issue for children.
All my children are now teens and ride good, Italian lightweight road bikes, but before they were 10 they rode what I gave them and that was a heavy bike.
Really.... Just buy a nice bike that fits nicely and make sure it's well maintained, the headset is smooth and tight enough, the tyres pumped and the bearings quiet. If two present themselves as being good, buy the lighter one. Beyond that, do not worry too uch about weight.
Do not make weight the main decider.
People can get a little precious about some technical issues, but there are bigger deals when learning to ride than a few kilos.